All other candidates remain in single digits in the latest survey, with Ben Carson at 6 percent, Jeb Bush at 3 percent, Mike Huckabee at 3 percent and Rand Paul at 2 percent.
Trump has been battling Cruz for the past two weeks for support in Iowa.
This week Trump unveiled an endorsement from former 2008 Republican vice presidential candidate Sarah Palin, who campaigned with him in Des Moines. Iowa Gov. Terry Branstad also called for the Texas senator to be defeated.
While Cruz was considered a month ago to be a favorite to win the Feb. 1 caucuses in Iowa, several recent polls have indicated a much tighter race between him and Trump there.
The pair have battled for the support of evangelical voters.
The CNN/ORC poll of 266 likely GOP caucus-goers was conducted Jan. 15–20 via telephones with a margin of error of 6 percentage points.
Three recent polls in the state have shown more malleability in the race between the two front-runners, with one finding Cruz with a 2-point edge and two others putting Trump up by either 1 point or 6 points.
Nearly half of respondents in the latest CNN/ORC poll, 49 percent, said they have "definitely decided" on a candidate, while 22 percent are leaning toward one and 28 percent are undecided.