Senate Majority Whip Richard J. Durbin (D., Ill.) Tuesday called on Rep. Joe Sestak, Pennsylvania's Democratic Senate nominee, to divulge details about his assertion that the White House had offered him a job in exchange for dropping his primary campaign.

"At some point, I think Congressman Sestak needs to make it clear what happened," Durbin told reporters on Capitol Hill.

Durbin, a White House ally who is close to President Obama, is the highest-profile Democrat to push Sestak to be more forthcoming on the issue. Last week, Sestak defeated five-term Sen. Arlen Specter for the party's nomination.

Sestak said in a February interview that the White House offered him a federal job last summer to forgo a challenge to Specter, who had Obama's backing after switching parties.

Since then, Sestak has declined to say what post was offered or who offered it - and he did so again Tuesday. White House officials, including political adviser David Axelrod and press secretary Robert Gibbs, have acknowledged that there were conversations with Sestak but say nothing "inappropriate" happened.

Offering a job to influence an election could in some circumstances be considered a violation of federal law, akin to a bribe. White House officials throughout history have used administration jobs as incentives but are usually careful to avoid a direct quid pro quo.

Durbin said Sestak had the responsibility to explain because "he raised the issue."

The Justice Department has rejected a GOP request for a special counsel to investigate the matter. Rep. Darrell Issa (R., Calif.), ranking minority member of the House Committee on Oversight and Government Reform, has demanded a criminal investigation.

"I think it brings back the whole Nixonian question of: It's not about what was done wrong, it's about the cover-up," Issa said Tuesday on Fox News Channel. "And right now there's a cover-up going on at the White House."

The White House did not respond to a request for comment.

Sestak said in a statement Tuesday afternoon, "Politics like this have too long dominated Washington and, for me, it's the people of Pennsylvania who matter."

Speculation has centered on the post of Navy secretary, since Sestak was a rear admiral who served for 31 years. But he has said the offer was made last July, a time when a new Navy secretary, former Mississippi Gov. Ray Mabus, was in place. Obama nominated Mabus on March 27, 2009, and the Senate confirmed him May 18. Specter became a Democrat on April 28.

Republicans were enjoying the Democrats' discomfort.

GOP national chairman Michael Steele said during a rally Tuesday in Blue Bell: "Don't you just love Congressman Sestak right now? Singing all kinds of tunes. White House is not happy."

Asked after the rally about the issue, Republican Senate nominee Pat Toomey said: "I do think it would be helpful if Joe would be more forthcoming and clear the air, but I'm not going to dwell on this."