(CNN) After a historic summit in Singapore, Republicans back home are warning President Donald Trump on Tuesday not to let the lure of a deal with North Korea cloud the realities of dealing with a dictator.

Congressional Republicans offered the President measured praise for his meeting with Kim Jong Un, with many saying they were hopeful that it could potentially lead to Pyongyang giving up its nuclear weapons. But they also warned that North Korean leaders have made promises before that they never followed through on — and numerous Republicans drew the line at the notion Trump raised that the US could ultimately withdraw its thousands of troops in South Korea.

Vice President Mike Pence met with senators on Capitol Hill on Tuesday, and Trump called into the meeting. Trump, speaking on Idaho Republican Sen. James Risch's iPhone, praised the summit and touted its success when speaking to GOP senators at lunch Tuesday, a source who attended told CNN.

Sen. John Barrasso, a Wyoming Republican, said the President sounded "confident and upbeat" in his call.

Sen. Roy Blunt, a Missouri Republican and a member of Senate GOP leadership, was cautious about the North Korea talks, saying before the meeting with other GOP senators, "We may have started down a path here that gets us to our goal, which is denuclearization."

But he added: "I'd be concerned about any discussion of removing troops from Korea. And I will look closely at what the discussion was about our joint operations -- our joint military operations."

Sen. Joni Ernst said she thought the summit was successful, but she also cautioned against withdrawing US troops.

"As far as moving troops off the Korean Peninsula, I think we need to stay there," the Iowa Republican said. "We don't just withdraw our troops, no."

The summit was just the latest example of Trump's ad hoc foreign policy, a meeting that under another president would likely have drawn the ire of foreign policy hawks back in Washington.

But instead, the President's party is trying to grapple with the North Korean reality, accentuating Trump's gesture as one of the only ways the US can tame a nuclear-armed enemy.

"It's not something we should celebrate. It's not a pretty sight, but it's the necessary part of a job to protect Americans," Sen. Tom Cotton, an Arkansas Republican, told radio host Hugh Hewitt on Tuesday.

Senate Foreign Relations Chairman Bob Corker, a Tennessee Republican, told reporters that he still has questions about what — if anything — was agreed to, and that he hoped to have Secretary of State Mike Pompeo brief him in short order.

"I think we've done that," Corker said when asked if he worried about legitimizing North Korea. "At this juncture, I don't think we know enough to challenge or celebrate"

The Republican message to Trump was clear: Verifying Kim's commitment to denuclearizing the Korean Peninsula is a must. While Trump told ABC News on Tuesday , "I think he trusts me, and I trust him," Republicans were skeptical.

"He's trying to be cordial, but I am a Ronald Reagan guy -- 'trust but verify,'" said Texas Sen. John Cornyn, the No. 2 Senate Republican.

"I think it's important that we don't lose sight of the fact that Kim Jong Un is a butcher and he is a butcher of his own people," said Sen. John Kennedy, a Louisiana Republican. "Trying to reason with someone like that, is like trying to hand feed a shark. Doesn't mean you can't do it, but you've got to do it very, very carefully."

In the joint statement released by the US and North Korea, Kim said he "reaffirmed his firm and unwavering commitment to complete denuclearization of the Korean Peninsula." But Republican senators said they were skeptical of Kim's promises, given North Korea's track record in past negotiations.

"I hope that he gives up his nuclear weapons in a verifiable way and it all works out," said Sen. Marco Rubio, a Florida Republican. "And the President has gone down that road and should be given a chance to succeed but I also think it's important for us to be cautious about what such a deal would be. It has to be real."

Senate Majority Leader Mitch McConnell also focused on the need for verifying that North Korea would give up its nukes.

"The next steps in negotiation will test whether we can get to verifiable deal which enhances the security of Northeast Asia, our allies, and of course the United States," McConnell said.

Democrats, meanwhile, slammed the summit as lacking an overall strategy and providing legitimacy to a brutal dictator.

"I have to be honest with you, this is the weakest statement I have ever seen come out of any engagement with North Korea, much less at the highest ranking of the President of the United States meeting with Kim Jong Un," said New Jersey Sen. Robert Menendez, the top Democrat on the Senate Foreign Relations Committee.

Senate Minority Leader Chuck Schumer said the meeting was "a welcome improvement" to "name-calling and saber-rattling," but argued that details coming out of the summit were lacking.

"It's imperative that we actually get action here, not just photo ops," Schumer said. "If the United States is unable to win concrete, lasting concessions from North Korea, the meeting alone will be a victory for Kim Jong Un and a defeat for President Trump."

Before departing Singapore Tuesday, Trump said that the US would stop military exercises with South Korea, which has long been a source of tension for Pyongyang. "Under the circumstances that we're negotiating a very comprehensive complete deal I think it's inappropriate to have war games," Trump said.

Republicans said they wanted more information about what Trump's military plans entailed.

While several lawmakers said they thought such a concession wouldn't harm the ability for the US military to respond in the Korean Peninsula, others pointed to it as another reason Congress should approve any deal with Pyongyang.

"I think it's unclear what he meant by that," Colorado GOP Sen. Cory Gardner said of military exercises. "I think it's important that we continue to engage in South Korea as we have been, from an exercise standpoint. But here's where I do think it's important: I think the President has talked about the good faith efforts and our progress being made towards denuclearization and it's clear our sanctions remain in place, that's why I strongly support maximum pressure."

Sen. Lindsey Graham, a Republican from South Carolina, said he's "fine" to put "war games" on hold — but "one thing he cannot entertain is withdrawing our troops as part of the deal."

Graham told CNN that that "something this big has to come to Congress" for approval, assuming there's a nuclear deal with North Korea. Graham warned North Korea: "If you try to play Trump or back out, there's going to be a war and nobody wants war. Look at what he said to Canada when he thinks he's wronged."