Donald Trump and Ted Cruz. AP The flap over presidential candidate and Sen. Ted Cruz's (R-Texas) Canadian birth could prove significant in the nation's first-voting state, according to a new survey from the Democratic firm Public Policy Polling.

The poll, which was shared with Business Insider ahead of its release Tuesday, found a tight Republican race in Iowa, which will hold the nation's first caucuses early next month. Real-estate magnate Donald Trump held a 28% to 26% lead over Cruz.

Trump has shone a light on Cruz's Canadian birth in recent days, claiming that Cruz would have a "problem" if he were to become the Republican nominee. Most legal experts believe Cruz's American mother is enough to qualify him for the presidency as a "natural-born citizen," but courts have never ruled on the issue.

The new survey found that only 46% of likely Iowa Republican caucus-goers realize that Cruz was born outside the US. (More Iowa Republicans believe Cruz was born in the US than President Barack Obama.)

But just 32% of GOP caucus-goers said that someone born outside the US should be allowed to serve as president, while 47% think such a person should not be able to serve as president.

Among the latter group of caucus-goers, Trump crushes Cruz by a 40% to 14% margin.

"Our poll findings show the Birther Issue could actually cost Cruz Iowa," Tom Jensen, Public Policy Polling's director, told Business Insider in an email.

Overall, when respondents were told that Cruz was born in Canada, the vast majority — 65% — said it wouldn't make a difference in whether they support him or not. But 24% said it makes them less likely to vote for Cruz. Jensen argued that in such a close race, "that could end up being a difference maker."

Trump began talking about the issue after being asked about it during a Washington Post interview last Monday, in which he responded by saying that it could be a "precarious" issue for Republicans. Since then, Trump has mentioned it frequently in stump speeches and media appearances.

During a New Hampshire rally on Monday, Trump argued that the Republican Party "just can't" nominate someone in Cruz's position.

"In the case of Ted, he has to figure it out," he said. "Now, I think he can go in for some kind of a judiciary proceeding. Look, it doesn't matter what he does. You can't have a nominee who's going to be subject to be thrown out as the nominee. You just can't do it."

The Public Policy Polling survey is one of several that has shown Trump closing the gap with his now-chief rival in Iowa, after Cruz surged ahead of Trump there recently. Trump also held a two-point lead over Cruz in a Quinnipiac University survey of the state released Monday.

Trump and Cruz are in a dominant position compared to the rest of the field. The PPP poll found Sen. Marco Rubio (R-Florida) in third place at 13%. Retired neurosurgeon Ben Carson was in fourth at 8%, while former Florida Gov. Jeb Bush (R) was in fifth place at 6%.

Trump does have another potential advantage over Cruz: Less than three weeks before the caucuses, 73% of his voters said they're "committed" to voting for him, the highest percentage of any Republican. Just 58% said the same about Cruz.

View the full poll below: