The leadership was declared vacant within minutes of the Liberal MPs gathering in Parliament House for their regular meeting on Tuesday morning, triggering a vote.
Some lawmakers including former Prime Minister Tony Abbott argue the government should be focusing on cutting electricity prices instead of cutting emissions.
The Daily Telegraph reports Mr Dutton is still considering a challenge to Mr Turnbull at the Liberal party room meeting this morning or that he may hold off to build more support before striking next month.
"In politics you have to focus on what you can deliver", Mr Turnbull told reporters on Monday.
Malcolm Turnbull staged a second U-turn over climate change in three days yesterday, aimed at saving his job as his Liberal Party slid further behind in the polls.
He also said every cabinet minister - including Dutton - supports the PM.
Tears as Korean families reunite for first time in decades
More than 57,000 South Korean survivors have registered for the family reunions, which usually end in painful farewells. Almost 20,000 people have participated in 20 rounds of face-to-face reunions held between the countries since 2000.
While several backers of Mr Dutton have admitted to a "wait and see approach", nine cabinet ministers were ready to get behind the Home Affairs Minister - half of cabinet's 18 Liberals, The Australian reported.
He appeared to have support from Dutton at the time, who tweeted: "In relation to media stories today, just to make it very clear, the Prime Minister has my support and I support the policies of the Government".
"Mr Turnbull is truly a white flag prime minister", Mr Shorten said.
The Telegraph says Abbott would return to "a prestigious portfolio" such as Home Affairs or Defence and would be "tasked with being the Dutton Government's chief attack dog" to campaign against opposition leader Bill Shorten.
While Coalition marginal seat holders have publicly supported Mr Turnbull, some privately lobbied in favour of Mr Dutton to improve the party's chances in Queensland.
Prime Minister Malcolm Turnbull will recontest his position, challenging his agitators to put up or shut up.