The Federal Deposit Insurance Corp.’s No. 2 official, Martin Gruenberg, has emerged as a top contender to succeed Sheila Bair as chairman, people familiar with the matter said, a move that could be part of a broader push by the White House to fill top bank-regulatory posts. Ms. Bair, a Republican who has led the agency since 2006, plans to step down when her term expires in a few months. Mr. Gruenberg has been the FDIC’s vice chairman since 2005, having served briefly as acting chairman before Ms. Bair took the helm in 2006.