According to White House staff, Mr. Biden is open to appointing Mr. Blum as an ambassador who is among the most desirable positions in any administration. After prioritizing the appointments of their West Wing and Cabinet staff, the President and his top advisors have only recently considered who to send overseas.

A potential ambassador, speaking on a sensitive subject on condition of anonymity, said Mr Biden himself wanted to think about the list of potential candidates and did not feel rushed.

There is, however, an increasing impatience among potential envoys. Former Senators, including some who have served in the Senate with Mr. Biden, are trying particularly hard to clarify and have noted how few in their ranks have joined the administration, according to a prominent Democrat who spoke to them .

The President, himself a former chairman of the Senate Foreign Relations Committee, is attempting a delicate balancing act: He rewards loyal donors and former colleagues without flooding the diplomatic corps with political representatives, as some of his staff do for former President Donald J. Trump had held.

Former senators who could be named ambassadors include Jeff Flake, the Arizona Republican who supported Mr. Biden; Joe Donnelly from Indiana; Heidi Heitkamp from North Dakota; Ken Salazar from 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 government but are waiting based on Mr Biden’s own deliberations and the narrow Democratic majority. For example, Nevada representative Dina Titus, an early Biden supporter, is hoping for a message, but currently there are three Democratic posts in the House where the party has a slim majority.

However, Mr Blum’s wish for a message could prove to be a consequence. California Governor Gavin Newsom promised in an interview with MSNBC’s Joy Reid Monday night that he would appoint a black woman to replace Ms. Feinstein. He admitted that he had “multiple names in mind” for a vacancy that didn’t exist.