Sestak, who defeated Specter in the Democratic primary last week, has said there was a job offer, but isn't providing details. The White House is mostly silent, saying nothing inappropriate was discussed.
The Delaware County Daily Times reported National Republican Senatorial Committee spokeswoman Amber Marchand said Monday Sestak "received" a bribe offer from the Obama administration.
Rep. Darrell Issa, R-Calif., called the alleged offer "an assault on our democracy" and "downright criminal," the Upper Darby Township, Pa., newspaper said.
Issa, a member of the House Committee on Oversight and Government Reform, called on the Justice Department to investigate whether a federal law barring federal employees from using "official authority for the purpose of interfering with, or affecting, the nomination or the election of any candidate" to the U.S. Senate and other seats was violated.
Sestak faces former U.S. Rep. Pat Toomey in the November general election.