Nothing makes friends like winning. It was not long ago, Mr. Sestak said Wednesday in an interview, that White House officials were so eager to muscle him from the race that they offered him a job if he would drop out.

Mr. Sestak remains mum on the details, except to say that it was a high-ranking post — secretary of the Navy has been mentioned as a possibility — and that it happened last summer. The White House, which had backed Mr. Specter, has denied the assertion.