It has been organised by Tory peer Lord Cruddas and former MEP David Campbell-Bannerman.
Lord Cruddas said: “The ousting of Boris Johnson as Prime Minister by a minority of MPs is deeply anti-democratic.
“It defies the will of the country and the Conservative Party members who elected him.
“It amounts to a coup. I am ashamed this can happen in Britain, the birthplace of modern democracy."
Prime Minister Boris Johnson departs 10 Downing Street, Westminster Dominic Lipinski
Liz Truss and Rishi Sunak have made it through to the final two in the Tory leadership race PA
Mr Johnson is also said to have sat down with Lord Cruddas of Shoreditch at Chequers on Friday.
And, the Telegraph reports that Mr Johnson "does not want to resign" as Prime Minister and wishes he could "wipe away" his departure.
The pair allegedly discussed the “bring back Boris” campaign for a second vote among the party's grassroots to confirm whether they think Mr Johnson should have to resign.
Lord Cruddas said Mr Johnson told him he was "enjoying following" the peer's petition and said he "wants to fight the next general election as leader of the Conservative Party".
He also told the newspaper: "There was no ambiguity in Boris’s views. He definitely does not want to resign.
"He wants to carry on and he believes that, with the membership behind him, he can."
Tory members have been signing the petition at a rate of nearly 2,000 a day since it was launched last week.
Anyone who signs the petition is asked to hand over their membership number when they sign up.
Downing Street has insisted Mr Johnson will step down once a new Conservative Party leader is in place.
A Number 10 spokeswoman, responding to comments from Lord Cruddas, said: “The Prime Minister has resigned as party leader and set out his intention to stand down as PM when the new leader is in place.”