Gillard refused to comment other than saying Fitzgibbon's words speak for themselves.
"I'll be happily leading Labor to the next election," she told reporters in Canberra.
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 Fitzgibbon's tweet was enough to settle the issue, 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 whether she had spoken to 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 yesterday.
"I don't want to dignify it with the attention it doesn't deserve."
Shortly before Fitzgibbon issued his tweet, Government frontbencher Greg Combet said 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," 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.
"I fully expect her [Gillard] to be there until the next election," she told Sky News.
"But if Labor changes its mind and changes its leader, then the Greens will continue to work with the Labor Party in the context of our agreement and we'd discuss that with any new leader."
She said she had a good working relationship with Gillard.
"I intend to continue to work in a very constructive way with her," Milne said.
After the 2010 election, Gillard and former Greens leader Bob Brown signed a deal for the Greens to support a minority Labor Government.
- AAP
