Editor's note: This story has been updated throughout.
At a rally here Wednesday evening, the U.S. senator from Texas named a time and a place for the showdown: 8 p.m. Saturday at Western Iowa Tech Community College in Sioux City.
"We have a venue, we have a time — all we're missing is a candidate," Cruz said, taunting his billionaire opponent as a "fragile soul" afraid of facing tough questions from Iowans.
As soon as Cruz's rally was over, the donors behind a pair of super PACs supporting him announced they would give $1.5 million to veterans' charities if Trump accepts the challenge.
“Senator Cruz and Mr. Trump both respect the veterans and hold them in the highest regard, but Senator Cruz respects the process, and we are calling on Mr. Trump to do the same, and debates are the purest form of democracy," read a statement from the donors, the Mercer and Neugebauer families. "Iowans — and Americans — deserve to hear from the frontrunners in this ‘two-man race’ one last time."
Cruz has been badgering Trump for a "mano a mano" debate since the billionaire announced Tuesday that he would not participate in the seventh GOP debate, which is being held Thursday in Des Moines. Trump, who's locked in a tight race with Cruz in this first-in-the-nation caucus state, complained that he would not get a fair shake at the debate because one of its moderators is Fox News host Megyn Kelly.
In a letter sent Wednesday to Trump, Cruz proposed a two-hour town hall-style event at the community college, which he said is already reserved for the occasion. Cruz said the meeting could be moderated by one of three conservative media personalities: Sean Hannity, Mark Levin or Rush Limbaugh.
Meanwhile, the super PACs, Keep the Promise I and Keep the Promise II, set four conditions for their $1.5 million donation: the debate must happen Sunday or earlier, it must last an hour, it must be between only Cruz and Trump and the candidates can choose the moderator on their own.
Trump so far has not agreed to anything with Cruz, whose campaign has started a full-court press to goad Trump into the showdown, launching a mini-website titled "DuckingDonald.com" and dispatching surrogates to call him chicken.
Cruz said Wednesday that Trump was making excuses in an attempt to avoid scrutiny of a record that does not match his rhetoric. Trump is not afraid of Kelly, Cruz told Iowans. "He's afraid of you."
The Trump campaign did not immediately respond to a request for comment on Cruz's invitation.