Grant Shapps, co-chair of the ruling Tory party between 2012 and 2015, claims having the backing of nearly 30 party MPs including former Cabinet ministers.
He plans to spend the weekend taking that number up to the required 48 MPs to write to the chairman of the Conservative Party's powerful 1922 Committee of backbench MPs to trigger a leadership contest within the party.
"I think it's time we actually tackle this issue of leadership and so do many colleagues. We wanted to present that to Theresa May privately. Now I'm afraid it's being done a bit more publicly," Shapps told the BBC.
Calls for the British premier's resignation have been growing ever since a doomed general election in June, which lost the Tories their overall majority in Parliament.
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Most recently, calls for May to step down have been revived as a result of a disastrous speech at the annual Conservative Party conference in Manchester.
Her key policy messages were all lost in a series of mishaps, including an interruption by a prankster, her own coughing fit and, to make matters worse, a faulty party message sign falling off on stage letter by letter.
He said the group of MPs supporting him included members across policy divides and covered Brexiteers and those who supported remaining in the European Union (EU).
But at the same time there are partycolleagues who have jumped to May's support and accused Shapps of stirring up trouble when an "overwhelming majority" want her to carry on as premier.
Gove's intervention came as UK home secretary Amber Rudd made a very public appeal in 'The Daily Telegraph' asking Mayto continue despite the "trio of mishaps" that blighted her conference speech.
Charles Walker, the vice-chairman of the influential Tory backbench 1922 Committee, also came out strongly against the Shapps-led coup attempt.
He said, "No 10 must be delighted that it's Grant Shapps leading this alleged coup. Grant has many talents but one thing he doesn't have is a following in the party, so really I think this is going to fizzle out to be perfectly honest".
Next week, on Wednesday, the 1922 Committee of backbench Tory MPs will meet for the first time since the party conference for what could be a crucial moment for the party's future and May's own leadership.