Australian Prime Minister Malcolm Turnbull reacts during a media conference at Parliament House in Canberra. Picture: Reuters

Canberra - Australian Prime Minister Malcolm Turnbull refused to step down on Thursday despite mounting pressure from Liberal party colleagues, but said he would allow a second leadership vote if a petition demanding it received enough signatures from party lawmakers.

At least 13 ministers had resigned from the government by midday Thursday, just two days after Turnbull narrowly won a leadership contest against former home affairs minister, Peter Dutton.

Turnbull told reporters on Thursday that if he received a petition signed by a majority of Liberal lawmakers asking for a leadership vote he would call a party room meeting for midday on Friday.

Turnbull, who has dropped in opinion polls recently, also said that he would not stand in another leadership contest.

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.

No Australian prime minister has served an entire term in office since 2007, due to a series of internal party coups. 

Turnbull himself ousted then prime minister Tony Abbott in 2015 while serving in his cabinet.

"Australians will be rightly appalled by what they're witnessing in their nation's parliament today and in the course of this week," Turnbull told reporters outside parliament on Thursday. 

"I think what we're witnessing at the moment is a very deliberate effort to pull the Liberal Party further to the right. And that's been stated by the number of people who have been involved in this," he continued.

He accused a minority in the party room and others outside parliament of trying "to bully, intimidate others into making this change of leadership that they're seeking."

Turnbull's time in office has been dogged by a schism between the moderate and conservative factions of his party. His attempts to appease the conservatives, including dumping a planned emissions reductions target earlier this week, have failed.

The government adjourned the lower house of parliament early on Thursday to focus on the leadership debacle, prompting opposition Labor leader Bill Shorten to tell Liberals in parliament, "Shame on you."

The petition against Turnbull would need a total of 43 signatures. He won Tuesday's internal party ballot against Dutton by just 48 votes to 35.

His opponents said Thursday they had enough support to oust him, but Australian media reported they did not have enough signatures on the petition.

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.

The prime minister also questioned Dutton's eligibility for parliament on Thursday.

The former minister's interest in two Brisbane childcare centres that may have received direct government subsidies since July this year could put him in breach of the constitution.

Dutton has said he is in the clear while Turnbull said he was waiting for the solicitor-general to report back on the issue on Friday morning.

Several senior ministers, including a top Turnbull ally, resigned on Thursday morning, an hour after Dutton announced that he had challenged Turnbull to a second leadership contest.

Finance Minister Mathias Cormann said the ministers had told Turnbull it was "in the best interests of the Liberal Party to help manage an orderly transition to a new leader."

"We are very conscious of the seriousness of the decision that we've made," added Cormann, who is known in Canberra as the "Kingmaker." 

Cormann voted for the more moderate Turnbull in Tuesday's internal vote but is also a close friend of Dutton.