Public dislike of Abbott is blamed in part for conservative governments suffering big election losses in Victoria state in November and Queensland state in January.
He has also been widely criticized for making Queen Elizabeth II's husband, Prince Philip, an Australian knight on Australia's national day last month.
Simpkins said in an email to party colleagues the knighthood for Prince Philip was "the final proof of a disconnection with the people."
"I think we must bring this to a head and test the support of the leadership in the party room," he wrote.
Bishop and Communication's Minister Malcolm Turnbull have been touted as potential replacements for Abbott.
While both have made public statements of support for Abbott, they can now sound out supporters from within government ranks now that the ballot is officially on the meeting agenda.
Abbott has warned his colleagues against such a challenge. He said Australians had voted out the chaotic and divided center-left Labor Party government in 2013 because it had changed its prime minister twice in four years.
MP Andrew Nikolic told colleagues in an email that the challenge was the "ill-disciplined and self-interested behaviours that the Australian people explicitly rejected in 2013."
"Your actions are disappointing and divisive," Nikolic told Simpkins in an email copied to other Liberal MPs. "You do not have my support for this."
Government MP Dennis Jensen, who like Simpkins is from Western Australia state, on Tuesday became the first to publicly state he had lost confidence in Abbott.
Colleague Sharman Stone said earlier Friday that the growing leadership crisis needed to be resolved next week when parliament sits for the first time this year.
"If Tony gets through this, we've got to get behind Tony," she said.
"If someone else does, that's our leader and we get behind that person and we diminish the prospect of having Labor back in because that would be totally catastrophic," she said.
- AP
