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Hillary Clinton holds a three-point edge over Senator Bernie Sanders in Iowa, a tightening of the race with roughly three weeks until voting begins, according to a new set of surveys of likely voters from NBC/The Wall Street Journal/Marist.

In the Republican contest, Senator Ted Cruz of Texas has 28 percent of support and Donald J. Trump has 24 percent in Iowa, but Mr. Trump maintains a commanding 16-point lead over Senator Marco Rubio of Florida, his closest competitor, in New Hampshire, the surveys show.

Mr. Rubio is in third place in Iowa at 13 percent, with Ben Carson at 11 percent. The other candidates are at 5 percent or less in the state.

In Iowa, Mrs. Clinton has 48 percent of support and Mr. Sanders has 45 percent as the competition between the two has become more contentious. Martin O’Malley, the third Democratic presidential candidate, has 5 percent in Iowa, meaning he will most likely qualify for the next presidential debate.

But in New Hampshire, the survey also found Mr. Sanders at 50 percent, to 46 percent for Mrs. Clinton.

That margin is smaller than a recent Fox News survey, in which Mr. Sanders led Mrs. Clinton by 13 points. An NBC survey last month showed Mr. Sanders ahead by nine points in New Hampshire.

Gov. Chris Christie of New Jersey, who is hoping for a strong showing in New Hampshire to vault him into contention for the nomination, has 12 percent in the state; Mr. Cruz has 10 percent; and Jeb Bush and Gov. John Kasich of Ohio each had 9 percent.

Mr. Rubio is in third place in Iowa at 13 percent, with Ben Carson at 11 percent. The other candidates are at 5 percent or less in the state.

The surveys also found Mr. Sanders, buoyed by the support of independent voters, outperforms Mrs. Clinton in hypothetical general-election matchups in both states among registered voters.

In New Hampshire, Mr. Sanders, who is from neighboring Vermont, beats all three, while in Iowa he tops Mr. Trump and Mr. Cruz and is tied with Mr. Rubio.

The surveys were conducted from Jan. 2 through Jan. 7. In Iowa, the survey of 456 likely Republican caucusgoers has a margin of sampling error of plus or minus 5 percentage points; the survey of 422 likely Democratic caucusgoers also has a margin of sampling error of plus or minus 5 percentage points.

In New Hampshire, the poll of 569 likely Republican primary voters has a margin of sampling error of plus or minus 4 percentage points. Of 425 likely Democratic primary voters, the margin of sampling error is 5 percentage points.