The North East Somerset MP has experienced a sudden upsurge in popularity since the General Election and he came second in a recent poll of party members on who should succeed Theresa May.
He joined Instagram in the run up to June’s election and has 40,000 followers – 22,000 more than the Prime Minister.
Some see the “young fogey” MP as a right-wing antidote to the popularity of Labour’s Jeremy Corbyn.
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.
Meanwhile, academic Ted Malloch told the Mail on Sunday that Mr Rees-Mogg had let slip his leadership ambitions during a private lunch.
“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 said.
“He did not mean now, but at some point in the future.”
And a friend of the MP quoted in the Sunday Times said Mr Rees-Mogg was “loyal to a fault” and would not do anything to “destabilise” Mrs May.
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”.
Responding to the reports, Mr Rees-Mogg played down such talk, but pointedly did not explicitly rule out standing for the leadership.
“I think if I threw my hat in the ring, my hat would be thrown back at me pretty quickly,” he said.