On one side, the hard-line Brexiteers are unhappy with the ‘soft’ deal beginning to emerge from the EU withdrawal negotiations.
On the other, Remainers are worried about the influence of hard Brexit supporters, including Boris Johnson, over Mrs May’s leadership decisions.
To trigger a leadership election, 15% of MPs (48 in the current crop) need to back a vote of no confidence in the Prime Minister.
Candidates then need to be nominated by two MPs to get onto the ballot paper. Tory MPs then vote using the first past the post system. When two candidates remain, members of the Conservative Party vote for their favourite, with the winner becoming the party leader.
But who would take over the tiller in the event of a leadership change?
These are some of the main contenders:
The Foreign Secretary has been tipped for the Tory leadership before, only withdrawing from the 2016 contest after former ally Michael Gove turned on him to launch his own bid. He’s popular with Conservative Party members, and could win support from hard Brexiteers.
However, the former London mayor is seen a something of a troublemaker: he has been accused of deliberately undermining Mrs May and is known to rile his fellow cabinet members. And his tendency to put his foot in it may well prove a stumbling block.
The ultra-traditional father of six and unlikely Instagram sensation has enjoyed an upsurge in popular support thanks to his unashamedly conservative straight talk and eccentric charm. He is staunchly pro-Brexit and lauded for his authenticity and intelligence.
But as an anti-abortion, anti-gay marriage, Eton-and-Oxford-educated former City of London millionaire, his far-right-of-centre social attitudes could get in the way of a serious bid to lead the party. What’s more, the backbencher has said that he doesn’t want the PM gig.
The environment secretary has one failed leadership bid under his belt after an eleventh-hour stab at success saw him scupper the chances of former ally Boris Johnson. He has remained popular with Conservative Party members, performing well in polls asking who the next party leader should be.
He’s pushed for a hard Brexit, earning popularity amongst leavers, and is seen to be doing well in his current cabinet post.
Amber Rudd has been generally well received among MPs and Tory members as home secretary. She has managed to keep a reasonable distance from cabinet infighting, helping bolster her image as a safe pair of hands.
However, the Hastings and Rye MP was a vocal Remainer in the run-up to the Brexit referendum, and a bid could well be blocked by Eurosceptic Tories. What’s more, she only won her seat by just over 300 votes in the most recent general election – a precarious position that wouldn’t help her chances.
The man at the helm of Britain’s exit negotiations with the European Union has been touted as a possible prime minister, backed by a support base of fellow pro-Leavers. The ambition is there, or at least has been in the past – he stood against David Cameron back in the 2005 leadership race and came second.
Tory party members also support the Brexit minister, although Jacob Rees-Mogg’s steep ascendance has eaten into his polling numbers. And with EU negotiations apparently tipping towards a softer Brexit, his stock amongst committed hard Brexit supporters is in jeopardy.
The Scottish Conservative leader has steered the party to its best election performance north of the border in more than 30 years. She’s also seen as likeable and charismatic – a welcome tonic for the Conservatives after the Maybot backlash.
But there’s a sticking point: Davidson isn’t an MP. In order to take the helm she would first need to run for a Westminster seat, something she’s said she has no interest in doing.
The former chief whip and notorious tarantula-owner has enjoyed a rapid rise through his party, culminating in his appointment as defence secretary last year. His upwards trajectory – and steely personal ambition – has put him in the running for a leadership bid.
Revelations about his personal life in recent weeks have damaged his chances however. Senior Tories have now warned that his hopes of becoming PM have been scuppered by damaging revelations about an extra-marital relationship with a former colleague during his time working at a fireplace manufacturer.