There has been much speculation that the MP for North East Somerset might consider the leadership, egged on by supporters who have dubbed themselves the "Moggmentum" movement.
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.
Jacob Rees-Mogg, the backbench Tory traditionalist, has dismissed reports that he could be the next Conservative party leader as products of the August "silly season".
Anglo-US academic Ted Malloch, meanwhile, claimed that Rees-Mogg had let slip his leadership ambitions during a private lunch.
However, the paper quoted a friend of the MP as saying: "Jacob is loyal to a fault and would never do anything to destabilise Theresa May".
He added: "He did not mean now, but at some point in the future".
They added however: "But should she stand aside. he is starting to look at some of his more ambitious peers and wonder what they have got that he hasn't".
"I think if I threw my hat in the ring, my hat would be thrown back at me pretty quickly", he said.
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.
Last week Mr Rees-Mogg was named as the second most popular choice to succeed Mrs May in a poll of party members by the ConservativeHome website.
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.
Some see the "young fogey" MP as a right-wing antidote to the popularity of Labour's Jeremy Corbyn.