The Health Secretary is said to be considering a run for Downing Street if Mrs May quits after Brexit is completed in 2019.
Mr Hunt is already sounding out senior colleagues and party donors about a potential run at Number 10, sources have said.
He campaigned for Remain in last year’s EU referendum, but is believed to have reassured MPs he has “changed his mind on Brexit”.
Tory insiders said he would be positioned as the alternative to Boris Johnson in any future leadership race.
The Foreign Secretary’s popularity has plunged after a string of recent gaffes, throwing his hopes of replacing Mrs May in doubt.
One source said: “It’s clear that Hunt is in the game and that more and more people are moving in his direction.
“He’s making very clear that he has changed his mind on Brexit. He’s been putting himself around to ministers, MPs and donors.
“He’s making clear that he’s someone with a business background, he’s pro-Leave, Boris is sinking and Michael Gove is non-viable.
“The Leavers might go for him as a way of stopping Amber Rudd and co.
“He’s getting some traction. A periscope has stuck up above the sea on this subject over the past month.”
Mr Hunt, who has also been touted as possible Chancellor, considered running against Mrs May in last year’s leadership campaign.
But he was advised against throwing his hat in the ring by allies, who feared his hopes of winning were unlikely.
They told him not stand while locked in a tense row with NHS junior doctors over controversial changes to their contracts.
But he has impressed colleagues recently by using TV and radio interviews to fiercely defend the Government’s record.
A top Department of Health source told the Sunday Times last night: “Jeremy is desperate for a new challenge.”
An aide to a pro-Brexit minister added: “Jeremy has been supporting Boris and Michael in cabinet, arguing that more preparations are needed for Brexit.
“At conference he was dropping into conversation that he would vote leave if there was another referendum.”
But one senior Conservative warned: “If the NHS is the Tory weak spot, which it is, having the most hated Health Secretary as the next leader – there lies electoral death and destruction.”
If he stands for leader, his biggest rivals would likely include backbencher Jacob Rees-Mogg and Scottish Tory chief Ruth Davidson.