Minister Simon Crean said government colleagues must sign a petition to force a ballot if Gillard refuses. Her office did not immediately comment, and the office of Kevin Rudd, her likely challenger, said he would issue a statement later.
Crean said he would nominate as deputy leader and wanted former prime minister Rudd to challenge for the top post. Crean is a senior minister for regional Australia, regional development and local government and arts.
Australian prime ministers are selected by lawmakers in the ruling party, but part of Rudd's appeal is opinion polling that shows Rudd would be a far more popular choice of the public.
He led Labor to victory at elections in 2007, then was deposed by his then-deputy Gillard in an internal party coup in 2010. He was roundly defeated in a ballot of Labor lawmakers last year, but recent reports have cited sources in saying he's now believed to have stronger support.
"Kevin Rudd in my view has no alternative but to stand for the leadership," Crean told reporters.
"He can't continue to play the game that says he's reluctant or he has to be drafted," he said.
"This is an issue that has to be resolved. There's too much at stake," Crean added.
Speculation has been growing that Gillard would be challenged before national elections in September. Opinion polls have indicated her party would be soundly defeated in the vote.