WASHINGTON — The Trump administration on Friday stepped up its effort to place a citizenship question on the 2020 U.S. Census, after the president said he was mulling an executive order to deal with the impasse and Justice Department lawyers told a federal judge they were still looking for a legal way forward.

The Supreme Court has called the administration's rationale for the question "contrived" and said the government could not go forward without a solid justification. On Friday, President Donald Trump and a key official mentioned other possible reasons for adding the query in statements that added to a weeklong swirl of contradictions and that could make the administration's case legal case even more difficult.

Government lawyers said in a filing Friday that the Justice and Commerce Departments had been “instructed to examine whether there is a path forward” for the question and that if one was found they would file a motion in the Supreme Court to try to get the question on the survey to be sent to every U.S. household.

Their filing came in a case before U.S. District Judge George Hazel in Maryland that poses the issue of whether the addition of the citizenship question would violate equal-protection guarantees and whether it is part of a conspiracy to drive down the count of minorities. He scheduled information gathering to begin immediately and conclude by Aug. 19, with any witnesses to testify in early September.

The government has begun printing the census forms without the question, and that process will continue, administration officials said.

Statements Friday from Trump and his acting U.S. Citizenship and Immigration Services director, Ken Cuccinelli, seemed to add confusion to why the government wants the addition.

The administration had said in multiple legal battles that the question was needed to get a better sense of the voting population to help enforce the Voting Rights Act. Opponents countered that the question could result in a severe undercount of immigrant communities.

But speaking to reporters at the White House Friday morning, Trump said the question was needed "for many reasons."

"Number one, you need it for Congress — you need it for Congress for districting," he said. "You need it for appropriations — where are the funds going? How many people are there? Are they citizens? Are they not citizens? You need it for many reasons."

Trump's statement could give additional heft to evidence discovered in May suggesting the administration worked with a Republican redistricting strategist who saw the question as a way to give Republicans and non-Hispanic whites an electoral advantage. Government officials had previously denied that adding the question had anything to do with the strategist or his analysis.

Appearing on Fox Business Network on Friday, Cuccinelli listed other justifications for the question: "Frankly, as part of the ongoing debate over how we deal financially and legally with the burden of those who are not here legally. That is a relevant issue."

Trump had raised the possibility that some kind of addendum could be printed separately after further litigation of the issue.

"We'll see what happens," Trump said. "We could start the printing now and maybe do an addendum after we get a positive decision. So we're working on a lot of things, including an executive order."

Whether an executive order or an addendum is feasible at this stage was not clear, and any shift almost certainly would carry extra costs.

Were Trump to issue an executive order, it is hard to predict what might happen: Much would depend on what it said, when it took effect, and what the Justice Department did to request court permission to deviate from its current course.

But it is likely those suing would return to the federal judges who have already blocked the citizenship question and either ask them to clarify that their injunctions apply to Trump's new executive order, or ask for new injunctions to yield the same effect.

If judges agreed, the administration once again would be stymied and forced to take the battle to higher courts. In the meantime, the administration will also have to work through discovery in the existing cases, potentially having to reveal more unflattering details about their handling of the citizenship question.

Thomas Wolf, counsel with the Democracy Program at the Brennan Center for Justice at the New York University School of Law, said there is "no path" forward for the government to add the question at this point.

"The court vacated the Voting Rights Act rationale as a contrived pretext," he said. "What they're trying to do now is the textbook definition of a pretext — telling the court, 'We plan to do this but we don't know why yet.' "

Anticipating a possible executive order, plaintiffs in a New York case that also challenges the citizenship question late Friday asked a federal judge in the Southern District of New York to permanently block the Commerce Department from adding it to the 2020 form. Judge Jesse Furman scheduled oral argument on that request for July 23.

Tacking on the citizenship question through an addendum is also fraught with legal and practical risks.

Census experts say that, among other concerns, an addendum would likely violate the bureau's strict rules on testing a question, which include considering how the placement of a question on the form affects respondents' likelihood of filling it out.

Terri Ann Lowenthal, a former staff director of the House census oversight subcommittee, said such a proposal is “neither operationally feasible nor scientifically sound, and it could not, at any rate, be pulled off in time for the 2020 census. The complexity of a census is as fine-grained as the weight of the packet that is mailed to households and the automated readability of forms. And the instructions for the original form would need to be revised to reference an ‘extra’ question on a separate page. The idea is little more than fantasy in the context of the current census.”

In litigation earlier this year, the government stressed that forms needed to go to the printer by July 1, prompting the Supreme Court to expedite its consideration of the question.

Trump's comments about finding an alternate route came as government lawyers scrambled to find a legal path to carry out the president's wishes despite their conclusions in recent days that no such avenue exists, according to people familiar with the matter.

“It’s kind of shocking that they still don’t know what they’re doing,” Thomas Saenz, president and general counsel of the Mexican American Legal Defense and Educational Fund, said. MALDEF is representing some of the plaintiffs in the case in Maryland. “We’re in this posture because they don’t know what the real plan is.”

Saenz derided the idea that an executive order could brush aside the 15 months of litigation that culminated in the high court's ruling.

"Executive orders do not override decisions of the Supreme Court," Saenz said. "Separation of powers remains, as it has been for over 200 years, a critical part of our constitutional scheme."

The government sought to hold off proceedings in the Maryland case, a request the plaintiffs said in their filing Friday, was "particularly inappropriate given Defendants' repeated representations to this Court and other courts, including the United States Supreme Court, that timing is of the essence."

The debate over adding the question had seemed settled after the Supreme Court ruled last week against the Trump administration. As late as Tuesday evening, Commerce Secretary Wilbur Ross, who oversees the census, said the administration was dropping its effort and was printing the census forms without it.

But Trump, in tweets Wednesday and Thursday, said he was not giving up.

The reversal came after Trump talked by phone with conservative allies who urged him not to give up the fight, according to a senior White House official and a Trump adviser, who both spoke on the condition of anonymity.

Trump told reporters Friday that the White House was surprised by the Supreme Court decision and that he found it "very shocking" that the citizenship question could not be included.

Trump said he believes the rationale provided by Ross "can be expanded very simply."

"He made a statement," Trump said of Ross. "He wrote something out. The judge didn't like it. I have a lot of respect for Justice Roberts. But he didn't like it, but he did say come back. Essentially, he said come back."

The Washington Post’s Colby Itkowitz and Matt Zapotosky contributed to this article.