Mr Abbott has also said he will not call an early election.
The speculation comes seven months after Mr Abbott survived a leadership challenge from his own party.
Since then, he has been unable to stem damaging leaks from his cabinet, including a list of cabinet ministers he was reportedly preparing to dump.
"My job is to govern effectively every day for three years then submit myself to the judgement of the people," Mr Abbott told reporters on Monday, referring to Australia's three-year federal election cycle.
"I am not going to get caught up in Canberra gossip," he said in reply to questions about leadership speculation.
Mr Abbott has struggled to push his promised reforms through the parliament because the coalition does not have a majority in the senate, Australia's upper house.
Popularity test
On Sunday, Liberal Party Whip Andrew Nikolic, strong supporter of the prime minister, asked senior cabinet minister Malcolm Turnbull to publicly disavow any plan to mount a leadership challenge, local media reported.
Mr Turnbull, who many see as a likely Liberal leader, has not commented.
Channel Nine news reported that Mr Abbott had told media mogul Rupert Murdoch he might hold an early national election after a by-election that will be held on Saturday, for the Liberal seat of Canning in Western Australia.
Opinion polls conducted by Australia's two major media groups, News Ltd and Fairfax, show a strong swing against Mr Abbott's Liberal Party in the seat, although it is expected to hold on to it.
A weak result in the seat would put further pressure on Mr Abbott's position.
The by-election was sparked by the death in July of the sitting Liberal MP Don Randall.
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