Heading into the Iowa caucuses, Trump had been projected to win, but Sen. Ted Cruz of Texas scored an upset. Establishment candidate Sen. Marco Rubio of Florida also did much better than expected, coming in a close third to Trump.
"It's a much different thing to go out and turn out voters on a cold night in Iowa or New Hampshire than it is to attract them to a Trump extravaganza where he provides entertainment for 90 minutes," Cantor told CNBC's "Squawk Box."
"This is again what I think lot of us have been saying about these polls, that when people get serious and start to vote, the question becomes who's going to be best put to be our commander in chief," said Cantor, a supporter of Jeb Bush.
The problem for the GOP is Trump is not going away anytime soon, said Robert Johnson, chairman of the RLJ Companies and longtime friend of the Clintons.
Trump commands 31 percent of support from likely Republican voters in New Hampshire, according to the most recent NBC News/Wall Street Journal/Marist poll, released Thursday. Cruz trails with 12 percent, followed by Rubio and Ohio Gov. John Kasich with 11 percent each and former Florida Gov. Bush with 8 percent.
"If [Trump] survives New Hampshire and rolls into South Carolina, he's right back where he is, and so the Republican establishment is now in this twin problem of trying to keep Cruz and Donald Trump out of the race," Johnson told "Squawk Box."
"I don't see how you have the bandwidth of trying to slay two dragons at the same time, and at the same time, pump up Marco and Jeb," he added. "You've got two guys that have to fall on their face almost simultaneously."
The question is whether outsiders Trump and Cruz can maintain their momentum, especially as Rubio surges, former Democratic Rep. Harold Ford Jr. told CNBC.
Rubio's message "is one that is American — pro-American, new American — and it looks like what the Republican Party wants to be going forward," said Ford, Morgan Stanley managing director.
Rubio finished with 23 percent of Iowa caucus-goers' support, well ahead of other establishment candidates such as Kasich, Bush, and New Jersey Gov. Chris Christie, all of whom attracted less than 3 percent.
Chris Krueger, Guggenheim Partners senior policy analyst, said he expected Rubio to take the nomination, and he only feels more strongly after Monday's caucuses. That nomination could come next month if he prevails in the key Ohio and Florida races on March 15, Krueger told CNBC's "Squawk on the Street."
However, if two candidates split the vote, he said, that could force a brokered Republican convention, which occurs when no candidate has enough delegates to win the nomination.
In the near term, Krueger expects a culling of the field following the New Hampshire primary.
Cantor noted that Iowa does not have the best track record in predicting the GOP nominee. Former Arkansas Gov. Mike Huckabee and former Pennsylvania Sen. Rick Santorum won in 2008 and 2012, respectively.
He maintained that Bush has captured momentum on the ground in New Hampshire and his numbers are on the rise.
However, the NBC/WSJ poll shows support among likely New Hampshire Republican voters for Bush slipped in January. He is in fifth behind both Rubio and Kasich in that poll, and he ranks fourth in an average of numerous polls crunched by RealClearPolitics.