Christopher Ruddy, chief executive of the conservative Web site, told The New York Times the site's readers "really love Trump" because they see him as not "owned by the Washington establishment [or] the media establishment."
The debate will be broadcast on ION Television Dec. 27 from Des Moines, Iowa, a week before the Iowa caucuses Jan. 3.
"I was asked to do it by a number of people, including some of my friends that are Republicans, strong Republicans," Trump said on Friday's edition of MSNBC's "NewsNation."
"I thought it would certainly be a little change of pace for Donald Trump, so I'll do something I haven't done before," Trump said. "I think I have a lot of good questions to ask. I feel very strongly about the country. So we'll see how it turns out. It could be the last debate for the primary in Iowa."
One candidate who will not participate in Trump's debate is former Utah Gov. Jon Huntsman.
"We have declined to participate in the 'Presidential Apprentice' Debate with The Donald," Huntsman spokesman Tim Miller said Friday, The Hill reported. "The Republican Party deserves a serious discussion of the issues so voters can choose a leader they trust to defeat President Obama and turn our economy around. We are confident that leader will be Jon Huntsman."
Miller said Huntsman will participate in a "Lincoln-Douglas"-style debate this month with fellow GOP presidential hopeful, former House Speaker Newt Gingrich.
"We look forward to discussing the issues that matter to the American people in a serious, substantive debate with Speaker Gingrich later this month," Miller said. "It is telling that Governor Romney has declined to participate because this substantive format makes it hard for him to hide from his record."
Newsmax began inviting candidates to the debate Friday afternoon, the Times said.
Trump this week said he would consider running as an independent candidate for president if the eventual Republican nominee is not to his liking. He flirted with a run for the GOP nomination this year but dropped out of the race, declining to make public his income tax returns.
He also considered running for president in 1987 and 1999, the Times noted.
President Obama likened Trump to a carnival barker for his repeated comments expressing doubt about Obama's place of birth.