Shapps also said this came from across the party, including those who wanted to leave the European Union, remain and those "somewhere in the middle" like himself.
'They are Remainers, they are Brexiteers, ' he told BBC Radio 4's Today programme.
She called the snap vote but ended up losing her parliamentary majority and her cabinet has been riven with divisions over Brexit strategy in recent weeks.
To trigger a formal leadership challenge, 48 Conservative lawmakers need to write to the chairman of the party's so-called 1922 Committee.
Mr Shapps said the list had been growing and was now up to around 30 names, although none of those are now in the Cabinet, and claimed it is "probably the time to have a leadership election".
The speech was also, at times, strikingly personal - May talked about her sorrow at not being able to have children and about her grandmother, who had been a domestic servant.
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"The truth is the overwhelming majority of Conservative MPs, the truth is the entirety of the Cabinet, the truth is the overwhelming majority of people, want the prime minister to concentrate on doing the job that 14 million people elected her to do earlier this year", said United Kingdom environment secretary Michael Gove, who had been one of the leadership contenders for the Conservatives alongside May when David Cameron exited 10, Downing Street after the Brexit vote last year.
The mock document showed the forged signature of current Foreign Secretary Boris Johnson, who many in the Conservative Party would like to be the next leader.
The British Prime Minister has insisted she is providing "calm leadership" with the "full support" of her Cabinet.
Mrs May retorted with a jibe aimed at the Labour Party leader, saying: "It's Jeremy Corbyn who should be given a P45". One senior lawmaker who believes May is deeply wounded said she would survive this crisis but the next one would end her leadership.
Shapps defended his decision to challenge May's leadership, saying: "We're not on the right path and the answer - after a frankly disastrous election - to have a new leader is hardly a radical thought".
A Scottish minister, Humza Yousaf said that people feeling sorry for a politician is the worst possible position for a politician, let alone a Prime Minister.
The Prime Minister's official Twitter account posted a picture of the text of her speech on a table surrounded by Strepsils cough candies, Olbas Oil and a bottle of Covonia.