WASHINGTON - House Speaker John A. Boehner on Tuesday rejected a new White House offer aimed at averting the tax increases and automatic spending cuts that are due to take effect in three weeks and countered with a plan of his own, an indication that the two sides are starting to negotiate.

President Obama and Boehner spoke by phone late Tuesday, the second time they had spoken in three days, after a face-to-face session Sunday at the White House. Neither side released details of Tuesday's conversation or the new offers.

One congressional source, speaking only on condition of anonymity to talk even a little about the secret offers, said the White House had made a new offer Monday seeking $1.4 trillion in new tax revenue over 10 years, a slightly smaller request than its earlier insistence that any package raise $1.6 trillion in new revenue over 10 years.

Apparently, the president did not come down far enough, nor did he provide the kinds of detailed spending cuts Boehner is seeking.

Republicans have offered to raise $800 billion in new revenue.

After receiving the offer, Boehner sent back his own.

The White House declined to comment on either proposal, both of which came after Obama and Boehner met Sunday for their first one-on-one meeting since the election last month.

White House press secretary Jay Carney said Tuesday that discussions had been cordial, a word Boehner also used. Carney added that "the lines of communication remain open," but he said he wouldn't provide updates as the negotiations continued.

Michael Steel, Boehner's spokesman, noted that the speaker had spoken before the House of Representatives earlier Tuesday and demanded that Obama provide specific spending reductions.

"As the speaker said today, we're still waiting for the White House to identify what spending cuts the president is willing to make as part of the balanced approach he promised the American people," Steel said. "The longer the White House slow-walks this process, the closer our economy gets to the fiscal cliff."

Carney said the president had offered a detailed proposal of spending cuts in 2011 and again this year.

"Now, it is entirely our expectation that Republicans may not agree with all of our spending cuts; Republicans may want to propose additional spending cuts," he said. "What we haven't seen from Republicans, to this day, is a single specific proposal on revenue, and in fact we've seen less specificity from Republicans on spending cuts than the president himself has proposed."

Republicans have accused Obama of playing a waiting game, hoping that enough Republicans will get nervous about the fiscal cliff and will agree to higher taxes without slicing into entitlement programs such as Medicare and Medicaid.

Several top Republicans have said they could back higher income-tax rates on the wealthy.