Following strong criticism of her exit deal, May used an interview on Sunday to emphasise the outline agreement on Britain's future relationship with the bloc was still being negotiated and would deliver on the 2016 Brexit vote.
According to reports around 25 have publicly submitted letters, although more could have sent them anonymously.
Sir Graham said the threshold had not yet been reached but if it was, he would act immediately to ensure a vote took place.
If 48 (15% of) Conservative MPs write such letters to the chair of the 1922 Committee, it means the party has lost faith in May's leadership and a new leadership contest must be triggered.
In a message to those plotting her downfall, including members of the European Research Group of Eurosceptic MPs, she said she had not considered quitting.
They specifically want to change the part regarding the arrangements for the border between Northern Ireland and Ireland which has been one of the main sticking points in talks with Brussels.
She will also talk about how the deal will allow Britain to control immigration, a key area of concern for many people who voted in the referendum.
Confronted in her constituency by Sky News, Mrs Leadsom said she was "absolutely determined to support the PM in getting the best possible deal for the United Kingdom as we leave the EU".
May will be traveling to Brussels to meet with EU leaders before an emergency European Council summit on November 25.
"She is tackling an issue of epic proportions, on which she can never please everyone, and she is doing her very best to find a way through", he said.
'It does deliver on the things that most people voted to leave the European Union for, such as control of our borders and laws.
After the Prime Minister unveiled her Brexit plan to cabinet ministers, resignations came thick and fast on Thursday, with Brexit secretary Dominic Raab, work and pensions secretary Esther McVey and more.
British newspapers reported that five senior pro-Brexit ministers were working to pressure May to change the deal, but May said she saw no alternative plan on the table.
"I still think a deal could be done", Raab told the BBC.
"It is very important to take the action now".
It also found a second referendum would be a closecall, with Remain beating Leave by 45pc to 43pc, with 10pc of respondents saying they had "had enough of Brexit and wouldn't vote".
Dutch Foreign Minister Stef Blok said Monday that "the withdrawal treaty is as good as it will get", while Luxembourg Foreign Minister Jean Asselborn also said there is "no better" Brexit deal for Britain.
"Instead of a system based on where a person is from, we will have one that is built around the talents and skills a person has to offer. What's the question going to be?"
The speech comes after Mrs May, who is due to travel to Brussels this week for talks with European Union leaders, used a Sunday interview to hit out at Tory rivals threatening to unseat her as party leader.
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