Cindy McCain, the widow of former Republican presidential candidate John McCain, has been floated as a possible selection in the media but UK diplomats play down its likelihood.
While she played a notable role in Mr Biden getting elected, with an endorsement followed by his victory in Mr McCain’s home state of Arizona, her appetite for the job is unclear.
Democratic senators such as Chris Coons, dubbed the ‘Biden whisperer’ for his closeness to the new president, have also been touted as possible picks.
Yet with the Democrats’ hold over the US Senate wafer-thin – they control the same number of seats as the Republicans, with vice president Kamala Harris getting the deciding vote on ties – such a path carries political risk.
Ted Kauffmen, one of Mr Biden’s oldest friends in politics who succeeded him as Delaware senator in 2009, ruled himself out of the running this week.
“No, no, no, no, no, no, no, no, no,” was Mr Kauffmen’s response when asked on Radio Four’s Today programme on Thursday if he would become the UK ambassador.
“My days of doing this are passed. I’ll be a helper, an adviser, but I’m 82-years-old,” Mr Kauffmen said. He demurred when told that was not much older than Mr Biden, 78.