Trump's urging came as he rejected the growing US government consensus that Russia was behind the hack of the Democratic National Committee -- a stunning display of a foreign power's meddling in a US presidential election.
A day later, Trump said he was being sarcastic with the comment.
But Trump refused to call on Russian President Vladimir Putin to stay out of the American politics and instead reasserted his desire to draw Russia and the US closer together as president -- building on a long-established track record of Trump's fondness for the autocratic Russian leader.
Here's a timeline of Trump's comments about Putin:
October 2007: Trump said Putin's doing a great job
"Look at Putin -- what he's doing with Russia -- I mean, you know, what's going on over there. I mean this guy has done -- whether you like him or don't like him -- he's doing a great job in rebuilding the image of Russia and also rebuilding Russia period," Trump told Larry King on CNN.
December 2011: Trump praised Putin's "intelligence" and "no-nonsense way" in his book "Time to Get Tough."
"Putin has big plans for Russia. He wants to edge out its neighbors so that Russia can dominate oil supplies to all of Europe," Trump said. "I respect Putin and Russians but cannot believe our leader (Obama) allows them to get away with so much...Hats off to the Russians."
June 2013: Trump wonders if Putin will be his "new best friend"
"Will he become my new best friend?" Trump asked of Putin in a tweet wondering whether Putin would attend the 2013 Miss Universe pageant Trump brought to Moscow.
October 2013: Trump says Putin is outsmarting the US
"I think he's done really a great job of outsmarting our country," Trump told Larry King after Putin successfully dissuaded the US from striking Syria by arranging with the US for the removal of Syria's chemical weapons.
July 31, 2015: Trump says they'd get along
"I think I'd get along very well with Vladimir Putin. I just think so," Trump said in one of his first comments about the Russian leader since launching his presidential bid last June.
Oct. 11, 2015: Trump says they had good ratings together
Asked on CBS' "Face the Nation" about similarities between him and Putin, Trump pointed to their appearance on same edition of "60 Minutes."
"I think the biggest thing we have is that we were on '60 Minutes' together and we had fantastic ratings. One of your best-rated shows in a long time," Trump joked. "So that was good, right? So we were stable mates."
Trump said he and Putin "are very different," but that they would "get along very well."
"I think that I would probably get along with him very well. And I don't think you'd be having the kind of problems that you're having right now," Trump said.
Nov. 10, 2015: Trump reiterates that he and Putin "were stablemates"
"I got to know him very well because we were both on '60 Minutes,' we were stablemates, and we did very well that night," Trump said, despite the fact that he and Putin had been interviewed in separate countries at different times for the same news program.
It's a comment Trump has repeatedly made at rallies.
Dec. 17, 2015: Trump returns Putin's praise
Donald Trump issued a statement after Putin praised the real estate mogul as a "talented person" and "the absolute leader of the presidential race."
"It is always a great honor to be so nicely complimented by a man so highly respected within his own country and beyond," Trump said in a statement. "I have always felt that Russia and the United States should be able to work well with each other towards defeating terrorism and restoring world peace, not to mention trade and all of the other benefits derived from mutual respect."
Trump took heat from his GOP rivals for the statement, but refused to back down.
Dec. 18, 2015: Trump defends against allegations Putin has ordered the killings of journalists
"He's running his country and at least he's a leader, unlike what we have in this country," Trump said on MSNBC's "Morning Joe." "I think our country does plenty of killing also."
Feb. 17: Trump says he'd be "crazy" to disavow Putin's praise
"I have no relationship with him other than he called me a genius. He said Donald trump is a genius and he is going to be the leader of the party and he's going to be the leader of the world or something," Trump said, embellishing Putin's praise.
"These characters that I'm running against said, 'We want you to disavow that statement.' I said what, he called me a genius, I'm going to disavow it? Are you crazy? Can you believe it? How stupid are they."
"And besides that wouldn't it be good if we actually got along with countries. Wouldn't it actually be a positive thing. I think I'd have a good relationship with Putin. I mean who knows," he continued.
April 28: Trump says maybe they'll get along
"Maybe we will, maybe we won't," Trump says when asked by Bill O'Reilly about whether he and Putin would have a good relationship.
"I'm saying that I'd possibly have a good relationship. He's been very nice to me," Trump said. "If we can make a great deal for our country and get along with Russia that would be a tremendous thing. I would love to try it."
July 28: Trump says he'd be firm with Putin
"I would treat Vladimir Putin firmly, but there's nothing I can think of that I'd rather do than have Russia friendly, as opposed to the way they are right now, so that we can go and knock out ISIS with other people," Trump said at a news conference.
But that wasn't Trump's only comment on Putin during the event.
He also refused to call on Putin to stay out of the election, "I'm not going to tell Putin what to do. Why would I tell him what to do?"
"Why do I have to get tough on Putin? I don't know anything other than that he doesn't respect our country," he continued.
Trump also predicted a better US-Russia relationship under his administration.
"President Trump would be so much better for US-Russian relations. It can't be worse," Trump said.
And of course, Trump said Putin would respect him more than Clinton, his Democratic rival who has been fiercely critical of Putin.
"I don't think he has any respect for Clinton. I think he respects me. I think it would be great to get along with him," Trump said.