"Australia no longer has a functioning government", he told parliament.
Australian lawmakers told reporters that party rebels were petitioning for another vote as early as Wednesday, though it appeared unlikely that the move would get enough support before Thursday.
Former Australian Home Affairs Minister Peter Dutton speaks to the media during a news conference at Parliament House in Canberra, Australia, August 21, 2018.
If Dutton's supporters succeed, Turnbull is likely to be ousted without completing three years in power, and whoever replaces him will become Australia's seventh prime minister in a decade.
"If the motion is carried, I will treat that as a vote of no confidence and I will not stand as a candidate in the ballot".
Turnbull also said he would leave parliament if he loses the leadership, which would trigger a by-election for his Sydney seat and could be problematic for the coalition government and its one-seat majority.
Mr Turnbull won the Tuesday contest only by seven votes.
Prime Minister Malcolm Turnbull has agreed to hold a leadership ballot at noon on Friday, but only after he receives a petition signed by a majority of Liberal members.
Turnbull's office had yet to respond publicly to Dutton's renewed leadership challenge or the latest spate of resignations.
Australian Prime Minister Malcolm Turnbull clung to power on Thursday as senior ministers deserted him.
"The government may adjourn the parliament, but they can not outrun the weight of failure of this government", he said.
The opposition Labor party has accused the government of being in "chaos".
Expectations that Turnbull would soon face another challenge, possibly within days, were stoked when eight Cabinet ministers backed Dutton.
The resignations were widely seen as pressure on Turnbull to call on a party meeting or to step down.
"There's far too much turmoil rather than good government". The names would provide proof that a majority of his government had abandoned him.
They have all been thrown out by their own parties. I'm not aware of anyone changing their vote.
"If this occurs, I will still attend National Party meetings if invited".
Australian media reported that Treasurer Scott Morrison and Foreign Minister Julie Bishop were considering contesting the leadership as well as Dutton, who has led a conservative rebellion against the more moderate Turnbull.
"I would put Tony Abbott into the cabinet as minister for indigenous affairs".
Many were surprised that he only lost by a 48-35 vote since he hadn't started canvassing support in earnest.
The balloting of party members who are not lawmakers takes a month, meaning Australians will never again wake to news that a Labor prime minister had been changed without warning.
Furious opposition lawmakers argued against the motion to adjourn the House of Representatives until September 10.
Turnbull said he asked Australia's most senior legal officer to provide advice on Dutton's eligibility. Some constitutional lawyers say there are areas of doubt.