"It is clear that neither party has earned the right to government in its own right,'' Gillard said.
"It's my intention to negotiate in good faith an effective agreement to form government."
However, Tony Abbott, the Liberal leader, said the loss of voter support for Labor that cost the government so many of its 83 seats showed that Australians wanted the government to change.
"It is historically unprecedented for a first-term government to receive the kind of rebuff that the ... government received yesterday,'' he said.
"It's certain that any Labor government emerging from yesterday will be chronically divided and dysfunctional."
While Abbott did not rule out winning the election, he confirmed that he also had preliminary discussions with the independents and the Greens.
Open to negotiations
Tanveer Ahmed, a columnist for the Sydney Morning Herald newspaper, told Al Jazeera that all parties have suggested they are open to negotiations.
"Minority parties and independents have received very strong backing," Ahmed said.
"One would expect the Greens to vote in line with Labor. It's a big win for rural Australians who have previously felt very disenfranchised.
"There is a broader impression that what Australians have voted for is chaos. I think the Australian government is headed for a period of significant uncertainty and possibly another election in the near future."
Tony Windsor, an independent, said he planned to talk with fellow independents
Bob Katter and Rob Oakeshott by telephone on Sunday to decide on issues
including whether to negotiate a power deal with the major parties as a group
All three were the only independents in the last parliament and all are former members of the Nationals party, which is a coalition partner of the Liberals. But all have said they are open to the prospect of supporting a Labor minority government.
"Whichever side it is, we need to have some stability and maintenance of stability so that the government can actually work,'' Windsor told the Australian Broadcasting Corporation [ABC] television.
"We might end up back at the polls."
Windsor was referring to the possibility of another election if a support pact could not be negotiated.
Bob Brown, the Greens party leader, said no agreement had been reached after a
"cordial'' conversation with Gillard, who was seeking the support of Adam Brandt, the newly elected Greens politician.
Brandt had previously stated his preference for a Labor government.
The Greens won a surge of support from former Labor voters after Labor shelved plans to charge major polluting industries for every tonne of carbon gas that they emit in a bid to curb Australia's greenhouse gas emissions.
The record support for the Greens has increased the party's senate seats from five to nine, giving them the leverage to become kingmaker in deciding which major party controls that chamber.
Andrew Wilkie, a former Greens member, the independent candidate contacted by Gillard, said on Sunday that he would not talk about which party he might support until his own seat was certain.