Gillard said a vote would be held on Monday when parliament resumes, after Rudd dramatically quit as foreign minister while in Washington on Wednesday amid reports she was planning to sack him for undermining the party.
"Australians are rightly sick of this and they want it brought to an end," Gillard told reporters of a battle for power that has been variously described by ministers as a "farce", "soap opera" and a "comedy".
She said she would again stand, and were she to lose would retire to the backbench and renounce future claims to the leadership, calling on Rudd to do the same.
"For far too long we have seen squabbling within the Labor party which has obscured the government's achievements and what we are doing to build a stronger and fairer Australia for the future," she said.
Australia's government has been torn by speculation about whether Rudd, whom Gillard brutally ousted as prime minister in mid-2010 but who remains popular with voters, would mount a bid to return to the top job.
He is yet to formalise his plans but in a clear pitch to caucus members, Rudd cited his record as prime minister, declaring the achievements of his government were formidable before boarding a plane back to Australia.
He laid out four key priorities under a government he would lead, including restoring business confidence, maintaining a strong manufacturing industry and continuing health and education reform.
In a broad hint that he wants his old job back, the 54-year-old said Gillard could not win the next elections due in 2013.
"I do not believe that Prime Minister Gillard can lead the Australian Labor Party to success in the next election," he said, adding that he was encouraged by the support he had received for a return to the top job.
"I have many more calls to make but their overall argument to me is that they regard me as the best prospect to lead the Australian Labor Party successfully at the next federal elections."
On current numbers, Rudd is likely to lose the ballot and be banished to the backbench. His other option is to vacate his seat, which would bring Labor an unwelcome by-election in his Queensland precinct of Griffith.
Local newspapers say around 49 MPs in the 103-member labor caucus are backing Gillard with 37 for Rudd and 17 undecided. A majority is required to win, although these numbers are shifting daily.
Gillard said her message to colleagues was that the party needed to unite after the vote "and get on with the job that Australians expect us to do".
"I expect to win," she said.
Since Rudd's resignation, senior ministers have been lining up to savage him in what has become a vitriolic and divisive showdown.
Treasurer Wayne Swan, who has cancelled his attendance at the G20 finance ministers' meeting in Mexico this weekend to help Gillard, called Rudd a "deeply flawed" man whose weaknesses were not seen in public.
But Rudd does have his backers, notably Senator Doug Cameron, who has been vocal in his support. Others reportedly include Immigration Minister Chris Bowen and Transport Minister Anthony Albanese.
The person most likely to benefit from the government's implosion is opposition leader Tony Abbott, who called the leadership battle "an embarrassment".
"It looks more and more obvious at the moment that this is a great country that is being damaged by a seriously bad government," he said.