Republicans increasingly think that Donald J. Trump has the best chance of winning the 2016 presidential election as their nominee as confidence fades in traditional politicians like Jeb Bush and Scott Walker, according to the latest New York Times/CBS News poll.

Although many Republicans have reservations about Mr. Trump and, as in past election years, could switch preferred candidates before voting begins this winter, the billionaire businessman appears to be gaining acceptance as a possible nominee.

The poll found that 39 percent of Republican primary and caucus voters viewed Mr. Trump as their best shot at winning the presidency, compared with 26 percent in a CBS survey in August. Only 15 percent said they would not back him as the party’s standard-bearer.

Ben Carson, another Republican with no history in the political establishment, is also gaining notable new support and is now running nearly even with Mr. Trump. Mr. Carson, a retired neurosurgeon who delivers his socially conservative message and outsider views in more measured tones than Mr. Trump, rose to 23 percent; he had 6 percent early last month before a widely praised performance in the first Republican debate. Mr. Trump draws 27 percent support in the new poll, compared with 24 percent last month.

The second Republican debate will take place Wednesday evening at the Ronald Reagan Presidential Library in Simi Valley, Calif.

Among Democrats, uneasiness with Hillary Rodham Clinton is growing and creating a possible opening for Vice President Joseph R. Biden Jr., as he considers making a late entry to seek the party’s nomination. Mrs. Clinton has lost the support of a majority of male Democratic voters and has also had a 7-point drop in support from Democratic women. Her lead over Senator Bernie Sanders of Vermont has shrunk from 41 percentage points to 20 points. Nearly six in 10 Democrats said they want to see Mr. Biden join the race, although it is not unusual for voters to seek additional choices early in a campaign.

The presidential race at this moment — a personality-driven free-for-all among 16 insiders and outsiders on the Republican side, an endless political therapy session about angst over Mrs. Clinton on the Democratic side — is so unsettled that national polls can be unreliable predictors this far out from actual voting for party nominees, which begins in February. At this stage of the presidential race eight years ago, Rudolph W. Giuliani and Fred D. Thompson led the Republican field, and Mrs. Clinton held an 18-point lead over then-Senator Barack Obama in a Times/CBS News poll.

The current New York Times/CBS News poll was conducted from Sept. 9 to 13 and has a margin of sampling error of plus or minus six percentage points for Republican and Democratic primary voters.