Republican presidential candidate Donald Trump speaks at the John Wayne Birthplace Museum on Tuesday in Winterset, Iowa. (Aaron P. Bernstein/Getty Images)

This post has been updated.

WINTERSET, Iowa -- As Donald Trump stood in front of a mannequin of western movie star John Wayne and accepted the endorsement of the late actor's daughter, speculation continued: What's the other big endorsement Trump will pick up on Tuesday?

There are rumors that later in the day Trump will announce the endorsement of Sarah Palin, the conservative icon who has championed the tea party movement. But Trump refused to reveal anything early.

"We'll talk about that later, but it's a very big event planned, and I think everyone is going to be very impressed," Trump said. "And nobody knows who it is, and that's fine."

Trump journeyed Tuesday morning to the John Wayne Birthplace Museum in Winterset, a small town about 30 miles southwest of Des Moines where the famed actor was born. Originally, Trump was supposed to rally with fans outside, but as the temperature hovered around 12 degrees and a steady snow fell, the event moved indoors. Trump commented that the stop was out of his way but that he was willing to do it because he loves Wayne, whom he met once, and hopes his appearance will bring the museum more visitors. Wayne's daughter Aissa Wayne endorsed Trump.

"We need someone like Mr. Trump with leadership qualities, someone with courage, someone that's strong -- like John Wayne," Aissa Wayne said. "And I'll tell you what, if John Wayne were around, he'd be standing right here instead of me."

Trump accepted the endorsement, saying he didn't realize it was coming, and told Wayne: "John Wayne would be very, very proud of you right now. I think? Maybe? Maybe he would say: 'What are you doing?'"

Not all members of the Wayne family agreed with the endorsement. Ethan Wayne, a son of the actor and president of John Wayne Enterprises, said in a statement that his sister organized the event independent of the Wayne family, John Wayne Enterprises or the John Wayne Cancer Foundation.

"No one can speak on behalf of John Wayne and neither the family nor the Foundation endorses candidates in his name," Ethan Wayne said in the statement.

Trump described John Wayne as "bigger than life" and said no actor today can match that. He then turned to a swarm of reporters and took some questions. The first one was about the bigger endorsement to come Tuesday evening -- and Trump clearly dodged it.

"We have a tremendous event planned in a little while -- that will be our third stop, this is our first stop," Trump said. "And, again, I just have to say I was just such a fan of John Wayne and the one meeting I had with him was just an amazing meeting... He was just an incredible guy. He said some things to me that were very special."

Trump was then asked if he was a fan of Palin. "I'm a big fan of Sarah Palin, yes," Trump said.

Is she going to endorse him?

"That I won't say," Trump said. "I am a big fan of Sarah Palin, but I'm not saying."

A Palin endorsement would be a major pick-up for Trump. The real estate mogul is trying to maintain his lead in Iowa ahead of the Feb. 1 caucuses as Sen. Ted Cruz (R-Tex.), a darling of the tea party, has been gaining support and major endorsements there.

Palin, the former governor of Alaska who was John McCain's running mate in 2008, no longer holds elected office, but she continues to be a popular spokeswoman for conservatives. She frequently offers political commentary on cable news channels, has starred in a reality television show and is a regular featured guest at conservative conventions. Her memoir, published in 2009, sold more than two million copies and was one of the fastest-selling political books in history.

In September, Palin was appeared alongside Trump and Cruz at a Capitol Hill rally against the Obama administration’s Iran deal. In December, Palin said she had narrowed her endorsement short list to those two, while her daughter Willow Palin backed Trump.

"I'm not going to pick one right now, but what a nice problem to have if it came down to Cruz and Trump," Palin told CNN in December. "That's a good problem for voters to have, because we know that, as you say, they are both strong and very decisive and someone who would take the initiative. That is what we need today, and both of those candidates would fit that bill."

On Tuesday, Reporters also had questions for Trump about Cruz's latest round of attacks on him, including the accusation that Trump lacks the temperament required for the presidency and for making military decisions.

"I think I have a great temperament -- in fact, if you noticed, I'm the only one that didn't want to go into Iraq," Trump said. "When you talk about temperament, Ted has got a rough temperament... You can't call people liars on the Senate floor when they are your leader. It's not a good thing to do if you want to sort of curry favor and get the positive votes later on down. So Ted is worried about his temperament, and people are talking about his temperament. I haven't talked about his temperament, but he's got to be careful, because his temperament has, you know, has been questioned a lot."

At one point during the event, which only lasted about 15 minutes, Trump called his Iowa director, Chuck Laudner, to the podium to defend the campaign's organization in the state ahead of the caucuses. Laudner and other state-level campaign staffers rarely are allowed to talk to the press, and Trump joked that if Laudner didn't do an adequate job that morning he would be fired.

Laudner said the campaign has held 13 training sessions in the past week for volunteers who want to help get more voters to the caucuses or speak on behalf of the campaign on caucus night.

"We feel fantastic about the ground game, and I know we go radio silent on those things, but there's nothing about this campaign that's like all the rest or any of those in the past," Laudner said. "We do things different, and we reach out to people that wouldn't normally be caught dead at caucus events, and so we feel really good about our chances. We feel really good about our reach, and I think you're going to have a surprise on caucus night."