Mr. Biden, according to White House aides, is open to appointing Mr. Blum to an ambassadorship, which is among the most coveted positions in any administration. After prioritizing the appointments of their West Wing staff and the cabinet, the president and his top advisers have only recently started considering whom to dispatch overseas.
One potential ambassador, speaking on the condition of anonymity to discuss a sensitive topic, said Mr. Biden himself wants to pore over the list of potential appointees and does not feel rushed.
There is, however, rising impatience among the would-be envoys. Former senators, including some who served in the Senate with Mr. Biden, are particularly eager to gain some clarity and have taken note of how few in their ranks have joined the administration to date, according to one prominent Democrat who has spoken to them.
The president, himself a former chairman of the Senate Foreign Relations Committee, is attempting a delicate balancing act: rewarding loyal donors and former colleagues without flooding the diplomatic corps with political appointees, as some of his associates thought former President Donald J. Trump had done.
Former senators who could be named as ambassadors include Jeff Flake, the Arizona Republican who backed Mr. Biden; Joe Donnelly of Indiana; Heidi Heitkamp of North Dakota; Ken Salazar of Colorado; and Christopher J. Dodd of Connecticut.
It’s not just former members of Congress looking for positions. A handful of current lawmakers are still hoping to join the administration but are waiting because of Mr. Biden’s own deliberations and the Democrats’ narrow majorities. For example, Representative Dina Titus of Nevada, an early Biden supporter, is hoping for an ambassadorship, but there are currently three Democratic vacancies in the House, where the party holds a slim majority.
Mr. Blum’s desire for an ambassadorship could prove consequential, though. Gov. Gavin Newsom of California, facing an increasingly likely recall threat and eager to energize his party’s base, pledged in an interview with MSNBC’s Joy Reid on Monday night that he would appoint a Black woman to replace Ms. Feinstein. He acknowledged he had “multiple names in mind” for a vacancy that does not exist.