In 2009, Turnbull, then leader of the Liberal Party in opposition, crossed the floor to vote with the Labor government in favor of an Emissions Trading Scheme, a move that ultimately lost him the party leadership.
Do you think Dutton should challenge Turnbull for leadership?
"Isn't it the case that when-ever it comes to a choice between his principles and his job, this Prime Minister stands for nothing at all", Mr Shorten said.
But Turnbull ousted Abbott as prime minister in 2015 in a leadership ballot of government lawmakers concerned by the government's poor opinion polling.
Asked further about the leadership, Mr Turnbull said: "I enjoy the confidence of the cabinet and of my party room".
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.
Peter Dutton is a senior conservative voice in the government who was identified as the major leadership rival to Mr Turnbull.More news: Federer and Djokovic to meet in final after Goffin withdrawal
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Mr Abbott said the internal war over energy was not a personality contest.
"It's not about him, it's not about me".
However, he declined to elaborate on the comment when quizzed by reporters yesterday and Ms Bishop later said she understood he had not said it.
But Treasurer Scott Morrison said was "fully supportive" of the Prime Minister.
Mr Turnbull had brought on the vote himself in a bid to resolve speculation that his leadership was under threat.
Monday's Fairfax/Ipsos poll shows the coalition's primary vote has dropped from 39 to 33 in a month, and Labor leading the coalition 55 per cent to 45 per cent on a two-party preferred basis.
Malcolm Turnbull is being assured he has the numbers to resist any immediate push from Peter Dutton after the Government dumped its carbon emissions reduction target to placate internal dissidents.