Democratic foreign policy leaders and congressional allies privately criticized the White House’s handling of the nominations, which some say follows a pattern of sloppy management of personnel selections. They worry that another retreat could embolden Republicans to challenge future nominations.

A top White House advisor said Friday that Obama was still considering Hagel, but that he was also looking at Michele Flournoy, who was Pentagon undersecretary for policy in Obama’s first term, and Ashton Carter, the current deputy secretary, the Defense Department’s second-ranking position.

The advisor said Obama hadn’t settled on Hagel and was not deterred by the criticism now. But with Rice out of the running for a Cabinet post, Obama may seek diversity among his new team and give greater weight to nominating Flournoy as the first woman to run the Pentagon.