Sir Tony Blair was tipped as a possible candidate. However, his spokesman said he was not in the running for the job. Boris Johnson was also named, but it is understood he is not considering running.
One Oxford insider suggested that Lady Angiolini KC, the principal of St Hugh’s College, could be nominated.
The Scottish lawyer became the first woman to serve as Lord Clerk Register of Scotland last year, after she was appointed by King Charles on the recommendation of the First Minister.
She was also appointed by Priti Patel, the former home secretary, as chairman of the inquiry into the abduction, rape and murder of Sarah Everard.
A serving don told The Telegraph that he would expect senior lawyers to be selected as candidates who have contributed significantly to public life.
“They need to have the stature to do the job as well as people skills,” he said.
‘Obvious-looking options’
Lee Phelps, of William Hill, said: “There are a couple of quite obvious-looking options for the Oxford Chancellor job following Chris Patten’s resignation, not least Rory Stewart, who tops the early betting at 5/6.
“Theresa May probably has about the right CV too. She’s a former PM with fairly moderate Conservative views and looks Stewart’s main rival in the race at 7/4.
“Fellow former leaders Sir Tony Blair and Boris Johnson could be in the shake-up and can be backed at 4/1 and 13/2, respectively.”
The contest comes as Cambridge also prepares for an election for its next chancellor, after Lord Sainsbury of Turville announced his intention to resign from the role.
The university plans to open up voting to graduates online. However, it has yet to set a timetable for an election and it has not been confirmed when Lord Sainsbury will stand down.