"That would be a crime to offer anybody a federal job," Rep. Darrell Issa, the top Republican on the House Oversight and Government Reform Committee, told Fox News on Friday.
For example, the California Republican said it would be a crime if he offered a staff job to anyone to help him win an election.
"It's the same for the executive branch," he said. "You can't promise ambassadorships to contributors and even worse, you cannot manipulate the races by saying we'll give you something else if you drop out. You can't do it."
Sestak, who is aggravating Democratic leaders by challenging Specter for the Senate nomination in Pennsylvania, said last month that the White House dangled a federal job in front of him last summer in an attempt to entice him to drop out of the state's Democratic primary.
Sestak has refused to elaborate on the circumstances but has acknowledged that the job was a high-ranking position.
Issa is seeking answers but the White House isn't talking. White House spokesman Robert Gibbs promised Fox News on Tuesday to provide answers. But since then, Gibbs has stonewalled reporters.
Issa wrote a letter to the White House counsel on Wednesday demanding information on Sestak's claim.
"While the White House may think this is politics as usual, what is spectacularly unusual is when a candidate – a U.S. Congressman no less - -freely acknowledges such a proposal," he wrote to White House counsel Robert Bauer. "Almost always candidates keep quiet about such deals, and for good reason – they are against the law."
Issa told Fox News that he is trying to give the White House an opportunity to tell its side of the story.
"Right now, they're doing the 'I won't confirm or deny,' and for us, it leaves two possibilities," Issa said. "One is the promise of transparency in this administration is just shot. The second one is even worse, which is either Sestak is lying or the administration has done something wrong and is covering it up and they should be the first to want to clear that up even if they're not wanting to support transparency as they said they would when they came to office."
Sestak stood by his claim this week.
"And I answered that yes, and I answered it honestly," Sestak told Fox News when asked if the White House offered him a job to not get in the primary.
"I would never get out for a deal," he said. "If I were to get out as I told this person, I would get out because it was the right thing to do. And the person responded, 'Yes, I knew you'd say that but…'"
Sestak declined to confirm that the position was for Navy secretary.
"As I said, there's nothing to be gained by focusing on this politics stuff," he said.
Specter told a local radio station Friday that Sestak could be committing a crime himself if the allegation is true and he hasn't reported it to the proper authorities.
"There's a crime called misprision of a felony," Specter said. "Misprision of a felony is when you don't report a crime. So you're getting into pretty deep areas here in these considerations."
Fox News' Neil Cavuto, Major Garrett and FoxNews.com's Stephen Clark contributed to this report.