JERSEY CITY, New Jersey (AP) -- Billionaire and Republican presidential front-runner Donald Trump returned to Fox News Tuesday morning for the first time since attacking the network's host during a debate last week and again refused to rule out running for president as an independent candidate.

While still considered a long-shot candidate for the Republican nomination in the 2016 election, Trump has been leading in the polls after stirring much controversy with contentious remarks about Mexican immigrants and lashing out at his opponents. Trump dominated the Republican party's first televised debate last week, where he sparred with the network's Megyn Kelly on the issue of his treatment of women.

In a phone interview with "Fox & Friends'' host Steve Doocy, Trump made only passing reference to his brouhaha with the network, one of the most powerful voices in Republican and conservative circles, saying that they'd "always been friends.''

He also refused once again to rule out a third-party run.

"I want to run as a Republican.  But I do want to keep that door open in case I don't get treated fairly,'' Trump said.

Trump and Fox News, have been in a bitter feud since the debate, in which he told Kelly that she has "blood coming out of her eyes, blood coming out of her wherever.''

The feud seemed to show signs of improvement Monday when the network's chairman, Roger Ailes, reached out to Trump directly, assuring him he'll be "treated fairly'' by the network, Trump tweeted Monday.

Kelly defended what she described as a "tough but fair question'' during the debate. "We agree to disagree,'' she said on her show Monday night.

"Trump, who is the front-runner, will not apologize. And I certainly will not apologize for doing good journalism,'' Kelly said.