Turnbull, who narrowly won a leadership vote on Tuesday against former home affairs minister Peter Dutton, said he would hold a second leadership vote on Friday if he received a letter signed by the majority of the ruling Liberal party. "And that's been stated by the number of people who have been involved in this", he continued.
Senator Mathias Cormann's resignation yesterday was called "the death knell" for Mr Turnbull.
Mr Morrison and Mr Dutton appear to have the most support. Turnbull said the meeting would proceed once the signatures had been verified.
Mr Dutton launched an unsuccessful leadership challenge on Tuesday, losing by a slim margin of 13 votes.
Other favorites include Treasurer Scott Morrison and Peter Dutton, a former cabinet minister.
Liberals elder statesman Jeff Kennett believes Ms Bishop has the best chance of helping the party win upcoming state elections in Victoria and NSW.
Mr Turnbull confirmed earlier on Friday he had received a petition showing this support.
He is yet to publicly respond to the latest threat to his rule, although broadcaster ABC reported that he turned down the demand for another party meeting during the phone call.
London house prices suffer largest drop in nine years
House prices in June rose by an annual 3.0 percent across the United Kingdom as a whole, the weakest increase since August 2013. Annual house price growth across the country slowed to its lowest level in almost five years in June, ONS figures also showed.
"I am optimistic about the potential of our people and will commit all my energy and experience to ensure the best years of our nation lie ahead of us".
"If I was putting my hat on, I would ensure that Julie Bishop was elected prime minister and Scott Morrison was deputy and retain the treasury for consistency's sake", Mr Kennett told Melbourne radio station 3AW. "I'm sorry. You deserve better than numerous things our Federal Parliament has served up to you for the past 10 years", wrote Nationals MP Darren Chester, whose party is in a coalition with the Liberals.
The turmoil came to a head after months of poor opinion polls and a revolt by fellow Liberal politicians on Monday against the prime minister's plans to embed carbon emissions targets in law at a time of soaring power prices.
Many Liberal MPs openly admit they don't have a clue about who will be the prime minister and leader of their party by early afternoon.
No Australian prime minister has lasted a full three-year term since John Howard lost power in 2007 after more than 11 years in office.
"Another day and another one of chaos from this government - a government that has effectively stopped governing because it's too busy fighting itself", said deputy opposition leader Tanya Plibersek. I'm sorry. You deserve better than numerous things our Federal Parliament has served up to you for the past 10 years.
In a move that could fuel those doubts, Attorney-General Christian Porter declared he would seek advice from the Commonwealth's top legal adviser, Stephen Donaghue QC, on whether Mr Dutton had a case to answer on his eligibility.
Many in Labor concluded that Rudd shouldn't have been dumped in the first place and the party changed its rules to make ousting a prime minister more hard.