Backbencher Luke Simpkins called the motion for Tuesday, saying it was “time to test the support of the leadership”.
Mr Abbott, who came to office promising stability, has faced growing questions about his position in recent weeks.
His party lost recent elections in Queensland and he was ridiculed for giving a knighthood to Prince Phillip.
Ms Bishop, whose position is also being challenged, had been seen as a potential leadership challenger.
In a statement on Friday she said there should be “support for current leadership” in Tuesday’s motion, known as a spill.
Social Services Minister Scott Morrison on Friday said he would not stand for election, Australia’s ABC News reported, while Communications Minister Malcolm Turnbull has not yet commented.
Mr Abbott’s leadership had been criticised before but his decision on Australia Day to award a knighthood to Prince Philip, the Duke of Edinburgh, was a game-changer.
Now, two backbenchers have come forward to lance what former Victorian Premier and Liberal heavyweight Jeff Kennett has likened to a festering boil of discontent with Mr Abbott.
The call for a party room spill on Tuesday by MPs Luke Simpkins and Don Randall has set the hares running. MPs are now furiously phoning each other to count the numbers for the pretenders.
Although Julie Bishop has supported Mr Abbott’s stand against the motion, that does not mean she is ruling herself out if the motion passes and goes to a leadership vote.
Leadership spills have failed in the past, however, and Mr Abbott could still survive to see another day.