The executive of the 1922 Committee is under pressure from Tory MPs to amend a rule that says a vote of confidence in the leader can only be held once a year. Ms May saw off an attempt to oust her in December, meaning another such vote cannot currently be held until December.
However, it appears to have agreed to hold off on any rule change until the vote on the Withdrawal Agreement Bill.
In a statement released after the meeting, Sir Graham said: "The prime minister is determined to secure our departure from the European Union and is devoting her efforts to securing the second reading of the Withdrawal Agreement Bill in the week commencing 3 June 2019 and the passage of that bill and the consequent departure of the United Kingdom from the European Union by the summer."
"We have agreed that she and I will meet following the second reading of the bill to agree a timetable for the election of a new leader of the Conservative and Unionist Party."
Labour has confirmed that it will vote against the bill, with shadow Brexit secretary Sir Keir Starmer telling the Commons: "I'd have thought it was patently clear that if the prime minister's deal is put for a fourth time, if it's allowed, it will fail just as it has failed three times already.
"But I want to make it clear that Labour opposes the idea of passing the Withdrawal Agreement Bill without an agreed deal - that would put the cart before the horse, and Labour will vote against at second reading on that basis."
It came as Boris Johnson confirmed that he will stand to succeed Ms May as Tory leader.
The former foreign secretary told a business event in Manchester: "I'm going to go for it. Of course I'm going to go for it."
As it happened...
Welcome to today's live coverage from Westminster.
Theresa May will hold talks with the executive of the 1922 Committee, which represents Tory MPs, this morning as she faces pressure to announce a date for her departure.
The prime minister has already said she will step down if her Brexit deal is approved by parliament but many Conservative MPs want her to announce a timetable for her resignation even if she does not get an exit plan through the Commons.
The 1922 executive is under pressure to change party rules to allow another vote of no confidence in the prime minister. It has so far refused to do so but it is unclear whether that will remain the case if Ms May refuses to give a date for standing down.
Liam Fox, the international trade secretary, has said he does not think that MPs would accept a permanent customs union with the EU.
He told BBC Radio 4's Today programme:
"I don't think parliament would actually accept the concept of a permanent customs union for a whole range of reasons that I've set out - and I don't think it would be acceptable inside the Conservative Party.
"Of course we do have a temporary customs union inside the implementation period, that is already accepted, but one of the reasons that we embarked on this particular process was so that we would be out of these arrangements by the time we got to the next general election."
Influential left-wing campaign group Momentum has urged Jeremy Corbyn to support a four-day working week
Speaking ahead of this morning's meeting of the 1922 Committee executive, the committee's treasurer, Sir Geoffrey Clifton-Brown, says Theresa May must set date for her departure or face being forced out.
He tells the Press Association:
"It would be infinitely preferable if she set a date rather than us force her out.
"It's better that she does it than we have a vote of confidence.
"What I would like to see is her set out a timetable to trigger a leadership contest."
Illegal immigration to the UK is likely to rise after Brexit, a new study has found
In the Commons, shadow Brexit secretary Sir Keir Starmer has just confirmed that Labour will vote against the Withdrawal Agreement Bill when it is put before the MPs at the start of June.
Yesterday Jeremy Corbyn's spokesman refused multiple times to rule out Labour abstaining on the bill.
But Sir Keir says:
"I'd have thought it's patently clear that if the prime minister's deal is put for a fourth time, it will fail just as it's failed three times already.
"I want to make it clear that Labour opposes the idea of passing the Withdrawal Agreement Bill without an agreement deal. That would put the cart before the horse and Labour will vote against at second reading on that basis."
He asks Brexit secretary Stephen Barclay "how on earth a bill to implement a deal that isn't before the House can pass in two weeks time" and suggests the decision to bring forward the bill is just designed to "keep the prime minister in office for another week".
Sir Richard Dearlove, the former head of MI6, has warned the government not to allow Chinese company Huawei a role in providing 5G infrastructure in the UK.
Ministers are weighing up the potential security risks of allowing the company access to UK networks, with reports suggesting that Theresa May is minded to do so.
In the foreword to a new report by the Henry Jackson Society think-tank, Sir Richard wrote:
"The fact that the British government now appears to have decided to place the development of some its most sensitive critical infrastructure in the hands of a company from the People's Republic of China (PRC) is deeply worrying.
"The PRC uses its sophisticated technical capabilities not only to control its own population (to an extreme and growing degree), but it also conducts remotely aggressive intelligence gathering operations on a global scale.
"No part of the communist Chinese state is ultimately able to operate free of the control exercised by its Communist Party leadership.
"To place the PRC in a potentially advantageous exploitative position in the UK's future telecommunications systems therefore is a risk, however remote it may seem at the moment, we simply do not need to take."
Labour MP Jess Phillips says she feels "sick" at Ukip MEP candidate Carl Benjamin's jokes about raping her.
Mr Benjamin, who had previously claimed he "wouldn't even rape" the Birmingham Yardley MP, defended his comments this morning, telling the BBC's Victoria Derbyshire Show:
"I'm aware of the politically correct narrative around this, but there's another narrative that I suppose we can call the non-politically correct one that I support.
"I think it's a lot more empowering to not be controlled by jokes. Survivors of sexual violence find what I've said empowering."
Labour has today announced plans to take the National Grid back into public ownership as part of proposals for a "green industrial revolution".
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