Just two days after an announcement that 1700 foreign workers would be brought in to work on a mining project in Western Australia, Ms Gillard said she was satisfied with Mr Fitzgibbon's words of support for her in a statement on Twitter on Sunday.
Union leaders have lashed out at the federal government's skilled migration plan that would allow the foreign workers to be employed by mining magnate Gina Rinehart at the Roy Hill Iron Ore project in the Pilbara.
Fairfax and News Limited papers said on Sunday that Mr Fitzgibbon, normally a Gillard supporter, was openly canvassing for votes to return Mr Rudd as prime minister.
Mr Fitzgibbon took to Twitter to say: "I thank my colleagues for the publicity but no one does more to support the PM and the government than me!"
Ms Gillard refused to comment other than saying Mr Fitzgibbon's words speak for themselves.
"I'll be happily leading Labor to the next election," she told reporters in Canberra.
Ms Gillard denied the government was divided and dysfunctional.
"We lead a government that is getting on with the job, most importantly keeping the economy strong," she said.
Asked whether Mr Fitzgibbon's tweet was enough to settle the issue, Ms Gillard said, "I think his words are clear."
"It's not the vehicle as to how they've been disseminated but what they say."
She would not be drawn on questions as to whether she had spoken to Mr Fitzgibbon since the reports emerged of his pitch for a leadership change.
Foreign Minister Bob Carr dismissed the reports of another possible leadership challenge as a "non-story".
"I don't think you can inflate a leadership challenge out of the news we're looking at this weekend," he told the Ten Network on Sunday. "I don't want to dignify it with the attention it doesn't deserve."
Shortly before Mr Fitzgibbon issued his tweet at 10.28 AEST, government frontbencher Greg Combet said Mr Fitzgibbon needed to answer questions about the newspaper reports.
He indicated he was surprised by the reports.
"I spent Friday with Joel Fitzgibbon and he didn't mention it to me," Mr Combet told ABC TV.
Greens leader Christine Milne said her party would continue to support the Labor government should Kevin Rudd be returned as prime minister.
