Sen. Christopher Dodd approached Federal Deposit Insurance Corp. Chairman Sheila Bair in recent days to gauge whether she would be interested in running the new consumer-protection agency, according to people familiar with the matter.

The chairman of the Senate Banking Committee's behind-the-scenes courtship of Ms. Bair suggests he is trying to find a nominee who might win favor in the Senate. It will be Mr. Dodd's job to move the nominee through a preliminary vote on his committee and then defend the person's record on the...