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Senator Ted Cruz and Donald J. Trump are essentially running neck and neck in Iowa less than three weeks before the state’s caucuses, according to a new poll released on Wednesday that suggests the two leading candidates could intensify their sniping in the homestretch of the race.

A Des Moines Register/Bloomberg Politics survey found that 25 percent of Iowa Republican voters support Mr. Cruz while 22 percent back Mr. Trump. Senator Marco Rubio and Ben Carson make up the second tier, with 12 percent and 11 percent, respectively, while the rest of the Republican field is languishing in single digits. The margin of error is plus or minus four percentage points.

The poll tracks with other recent surveys that indicate a tight race in Iowa, which begins the presidential nominating process with its caucuses on Feb. 1. Mr. Cruz, who has been consolidating the support of evangelical Christians there in recent months, has seen his support soften a bit since December as Mr. Trump has intensified his attacks.

The billionaire developer has raised questions about Mr. Cruz’s eligibility to be president because he was born in Canada and has questioned the Texas senator’s standing with evangelicals by noting his Cuban heritage. After months of trying to embrace Mr. Trump and his supporters, Mr. Cruz has finally started to hit back, mocking his knowledge of foreign policy and accusing him of being someone that “embodies New York values.”

The poll released on Wednesday found that most Iowans are not overly concerned with Mr. Cruz’s Canadian roots and that they find his strong Christian values to be very appealing. Mr. Trump is seen as the candidate more likely to disrupt the way government works.

The two candidates are spending significant time and resources in Iowa ahead of the caucuses. At a rally on Tuesday night, Mr. Trump joked that he might even buy real estate in the state someday.

“I love Iowa,” he said. “I think I’m gonna buy a farm and settle down out here.”