Speaking at an event Tuesday outside Philadelphia, Pat Toomey said he prefers to focus on issues but "it would be helpful" if Sestak cleared the air about the job offer.
Sestak and the White House have declined to discuss the offer in detail, but Sestak has said he was offered a job to avoid challenging Specter.
Sestak defeated Specter, a longtime Republican who switched parties last year, in last week's primary.
Toomey spoke at an appearance with Republican National Committee Chairman Michael Steele in Blue Bell.
Sestak campaign spokesman Jonathon Dworkin says his candidate's opponents are spending too much time trying to create a distraction.
