He resigned this week, blaming Ms Gillard, who replaced him as prime minister after a party coup in June 2010. She subsequently scraped home in a general election to form a minority government.
“It’s no secret our government has a lot to do if it is to regain the confidence of the people,” Mr Rudd said in Brisbane. “Starting on Monday, I’m going to start restoring that trust.
“I want to finish the job the Australian people elected me to do when I was prime minister.”
Mr Rudd quit as foreign minister during a visit to Washington, saying he could not continue in his role without the support of the prime minister. Ms Gillard then announced a party leadership election for Monday.
The government could fall if Mr Rudd wins because Labour’s single-seat majority depends on a coalition with two independents and a Green Party member.
Early elections would be held if neither Labour nor the conservatives can gain a majority.
Ms Gillard said: “The choice on Monday is a choice as to who has got the character, the temperament, the strength to deliver on behalf of the Australian people.”