It adds: "You admitted in the House that you are running from a heavy defeat on your deal in the House of Commons".
Many supporters of Brexit say May's deal fails to deliver on the clean break with the bloc that they want.
While Conservative former Cabinet minister Owen Paterson previously said he did not think his letter of no confidence in the Prime Minister had reached the mark.
"She's not the person to see Brexit through", he said.
"That must start here in Westminster with politicians on all sides coming together and acting in the national interest".
"And a leadership election would not change the fundamentals of the negotiation, or the parliamentary arithmetic".
After the meeting Rutte described the talks as "useful dialogue" but he has previously said "if there are people in Britain who think they can get a better deal, then they are wrong".
In a statement, the chairman of the 1922 Committee of Conservative MPs, Sir Graham Brady, said: "The threshold of 15% of the parliamentary party seeking a vote of confidence in the leader of the Conservative Party has been exceeded".
To survive the vote, May needs to get over half of the registered votes cast by the party's 315 MPs.
A ballot will be held between 1800 and 2000 GMT in a room at the House of Commons and an announcement made as soon as possible afterwards, he said.
If May loses the vote, with at least 148 members of her party casting a vote of no-confidence, she will be the second Conservative prime minister toppled by the tumultuous Brexit process.
Mrs May is now immune from a leadership challenge for a year. But if she loses she can not stand again and the party will choose a replacement.
These two candidates are then put to a postal ballot of the wider Conservative Party membership.
Earlier, Irish Prime Minister Leo Varadkar said the European Union could clarify what's in the withdrawal agreement but the substance of the deal will not be changed. All other Conservative lawmakers can run.
She said the deeper question was whether lawmakers wanted to deliver on the people's will from the 2016 referendum, or open up divisions with another national vote.
Cabinet minister Chris Grayling said: "This has hardly been a joyful day for the Conservative Party but it has voted very comfortably that it wants her to stay, wants her to take us through Brexit".
With this in mind, Work and Pensions Secretary Amber Rudd told Sky News: "I hope my colleagues will take a careful look at who's leading us, what the choices are and what the alternative is if they don't support it".
But some ministers expressed support for her with Home Secretary Sajid Javid saying a leadership contest was the last thing Britain needed.
But her spokesman was keen to stress: "This vote isn't about who leads the party into the next election, it's about whether it makes sense to change leader at this point in the Brexit negotiations".
Foreign Secretary Jeremy Hunt said May was "the best person to make sure we actually leave the European Union on March 29", while Home Secretary Sajid Javid said a leadership contest would be seen as "self-indulgent and wrong".
But Penny Mordaunt, Portsmouth North MP and worldwide development secretary, posted on Twitter and said: 'The Prime Minister has my full support, not least because she has always done what she firmly believes is in the national interest.