The Old Etonian, who recently announced the birth of his sixth child - Sixtus Dominic Boniface Christopher - is reportedly "sounding out" friends about whether he should go for it, according to The Sunday Times.
Tory MP Jacob Rees-Mogg has distanced himself from suggestions that he is considering a bid for the Conservative leadership.
"I was at a lunch with Jacob very recently and he indicated he would like to be considered for the leadership when the time comes", Professor Malloch, who is close to United States president Donald Trump, told The Mail on Sunday.
"He did not mean now, but at some point in the future". In the event Theresa May steps aside, one friend said: "He is starting to look at some of his more ambitious peers and wonder what they have got that he hasn't".
In an interview with The Sunday Times, Rees-Mogg, 48, sidestepped questions about his ambition, saying: "I think if I threw my hat into the ring, my hat would be thrown back at me pretty quickly".
He joined Instagram in the run up to June's election and has 40,000 followers - 22,000 more than the Prime Minister.
With his double-breasted suits, colourful vocabulary and confident media appearances, father of six Rees-Mogg, whose nickname is "the Honourable Member for the early 20th Century", has attracted a large online following.
But on Broadcasting House, and in comments to the two newspapers that covered his leadership prospects, Rees-Mogg declined to rule out standing as a candidate in the Tory leadership contest widely expected before the next general election.
But last week a poll of Conservative members for the ConservativeHome website found he was in second place, behind David Davis, the Brexit secretary, as their preferred next party leader. "It would be the end".