Former Liberal Party leader and Communications Minister Malcolm Turnbull said he asked the prime minister to open up the party's leadership to an internal vote, as the two-year-old conservative coalition government struggles in opinion polls.
It was not immediately clear if Abbott will agree, but he is now under intense pressure to prove he has the support of his colleagues.
"Ultimately, the prime minister has not been capable of providing the economic leadership our nation needs," Turnbull told reporters. "He has not been capable of providing the economic confidence that business needs."
Nick Economou, a Monash University political scientist, was not prepared to predict an outcome.
More From This Section
"The Liberals have done enormous damage to themselves, regardless of the outcome," Economou said. "I find it hard to believe that someone would move on the leadership unless they were absolutely confident of their numbers."
The possible leadership vote continues an extraordinarily volatile period in Australian federal politics, especially as the Liberals were elected in 2013 as a stable alternative to the then Labor government.
Before Rudd was elected in 2007, John Howard was in power for almost 12 years.
The government has trailed the opposition in a range of opinion polls since April last year. Abbott survived a leadership challenge from within his party in February that was prompted by those polls and what some say were questionable judgments he made.
Turnbull, a 60-year-old former lawyer and merchant banker known for his moderate views, has long been considered Abbott's chief rival. Turnbull was opposition leader for two years before he lost a party-room ballot by a single vote to Abbott in 2009. His downfall was his belief that Australia should make polluters pay for their greenhouse gas emissions.
Opinion polls show that Turnbull is more popular than Abbott, but many of those who prefer him vote for the center-left Labor Party.