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Hillary Rodham Clinton’s national lead over her rivals for the Democratic nomination has shrunk, and she is in a statistical tie with leading Republicans in head-to-head surveys, a new CNN/ORC poll released Thursday evening showed.

Mrs. Clinton is getting support from 37 percent of Democrats and Democratic-leaning voters, with Senator Bernie Sanders, the Vermont independent who is running in the Democratic contest, getting 27 percent, the survey found. Vice President Joseph R. Biden Jr., who is considering a run of his own and who has been included in several recent public surveys, received 20 percent support.

Martin O’Malley, the former Maryland governor, is at 3 percent, while Jim Webb, the former Virginia senator, received 2 percent, and Lincoln Chafee, the former governor of Rhode Island, is at less than 1 percent.

The new poll numbers come after a rough summer for Mrs. Clinton, who has been trying to move past calls for her to better explain her use of a private email server while she was secretary of state.

Mrs. Clinton’s share of support has shrunk 10 percentage points since last month in the survey. Among the striking numbers in the poll is that about 43 percent of Democrats and Democratic-leaning voters said they were excited about her candidacy, down from 60 percent when she announced she was running in April.

Yet Mr. Biden and Mr. Sanders are not stirring excitement among the Democratic faithful either — just 37 percent said they would be enthusiastic about a Biden candidacy, with 31 percent feeling that way about Mr. Sanders. And 30 percent said they would be dissatisfied should Mr. Sanders become the Democratic standard-bearer.

When Mrs. Clinton is pitted against each of the top three Republican candidates, Ben Carson, the retired neurosurgeon, leads 51 percent to 46 percent among registered voters; Jeb Bush leads 49 percent to 47 percent; and she and Donald J. Trump, who is leading most polls of the Republican field, are tied at 48 percent. All of those numbers are within the margin of sampling error.

The poll was taken from Sept. 4 through Sept. 8 and questioned 1,012 adults, including 395 people who identified themselves as Democrats or Democratic-leaning independents. The overall survey had a margin of sampling error of plus or minus 3 points, and of plus or minus 5 percentage points for the Democrats.