A spill could happen as soon as Tuesday morning when the Liberals hold their regular party room meeting, but is more likely later in the week, sources have told AAP.
Backbenchers Jane Hume and Sarah Henderson are among those who have questioned Ms Ley's leadership during a day of hand-wringing and speculation.
"I'm really tired of gallows humour, because that's all we've got left right now," Senator Hume told reporters in Canberra.
"I don't know what the solution is, and I don't know who the solution is, but what I do know is that more of the same simply isn't good enough."
The latest Newspoll, conducted during the most recent break-up between the Liberals and Nationals, puts the coalition's first-preference vote at just 18 per cent, eclipsed by Pauline Hanson's One Nation on 27 per cent.
More than 60 per cent of people surveyed were dissatisfied with Ms Ley's performance, making her the most unpopular major party leader in at least two decades.
Senator Henderson said the low primary vote was a crisis which meant her party was no longer a viable opposition and change was needed this week.
"We are in a dire situation," she told reporters in Canberra.
Liberal MPs will attend a party room meeting on Tuesday morning, where sources indicated a spill was possible.
But senators are stuck in committee hearings from Monday to Thursday, meaning a special meeting is likely to be called later in the week, allowing a challenge to take place with all members present.
It's unclear whether Mr Taylor has the numbers to win a leadership challenge - while his camp insists he does, some moderates privately claim his victory is not a done deal.
Ms Ley remained defiant on Monday, insisting she would remain as leader.
"I've been elected by my party room. I'm up for the job, we're up for the job, and we know that we have to hold this government to account," she told Nine's Today program.
Asked if his leader retained the support of her colleagues, one of Ms Ley's key factional lieutenants, Alex Hawke, said "obviously she does" and dismissed talk of a spill as "feverish speculation".
Shadow attorney-general Andrew Wallace called for cooler heads and warned his colleagues against overthrowing their first female leader.
"Were she to be deposed, I can almost guarantee you there'll be people out there saying 'you guys have just politically assassinated the first female leader of the Liberal Party'," he told reporters.
"This is a time to rebuild the coalition. This is a time for sensible voices."
Fellow Liberal senator James McGrath said he was angry at how far voter support had slipped.
"I'm not going to sprinkle gold dust on a cow pat," he told ABC Radio.
 "The polling is dire. It is horrible. It is terrible."
Recent opinion polling by Redbridge and DemosAU has shown similar results to Newspoll, with voters putting One Nation ahead or on par with the coalition.
Mr Taylor failed to put rumours of a spill to bed on Friday, telling Sydney radio 2GB he did have leadership ambitions.
Pressed on whether Ms Ley would still be leader in a week, he said a coup was not in the works but conceded he had been having conversations with his colleagues about the party's future.
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