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Donald Trump greeted supporters after his rally in Dallas on Monday.Credit Cooper Neill for The New York Times

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 Republicans have reservations about Mr. Trump and voters often end up embracing candidates who lagged in early polls, 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 be 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 seven-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 wanted to see Mr. Biden join the race, although it is not unusual for voters to seek additional choices early on.

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 voting for party nominees, which begins in February. At this stage 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.

Question: Which one of these candidates would you like to see your party nominate for president in 2016?

Change in support, from August to September, among Republican and Democratic primary voters. Those polling 1 percent or less in both months are not included.

 

Republicans gaining

or unchanged

Question: Which one of these candidates would you like to see your party nominate for president in 2016?

Change in support, from August to September, among Republican and Democratic primary voters.

 

Those polling 1 percent or less in both months are not included.

Republicans gaining

or unchanged

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.

The establishment candidates are battling fierce headwinds from a party electorate that vastly prefers a nominee from the business or private sector rather than a traditional politician, by 48 percent to 9 percent. Mr. Bush, a former Florida governor, and Mr. Walker, the governor of Wisconsin, have lost the most support by far since delivering what many analysts called lackluster performances in the August debate. Eleven percent of Republicans viewed Mr. Bush as best positioned to win in the general election, compared with 23 percent in August; only 2 percent thought Mr. Walker was most likely to win, down from 8 percent.

Carolyn Dillard, a 55-year-old Republican who is moving from Virginia to Florida, said in a follow-up interview that she was supporting Mr. Trump because he was “not your typical politician who says everything that is politically correct.” She credited him with targeting the national debt and illegal immigration.

Trust and Leadership

Primary voters from each party rate top candidates on two questions.

Do you think the candidate is

honest and trustworthy?

Biden

Carson

Bush

Clinton

Sanders

Trump

Do you think the candidate has strong

qualities of leadership?

Clinton

Trump

Biden

Carson

Bush

Sanders

Is your mind made up or is it too early

to say for sure?

Do you think the candidate is honest and trustworthy?

Joseph R. Biden Jr.

Ben Carson

Jeb Bush

Hillary Rodham Clinton

Bernie Sanders

Donald J. Trump

Do you think the candidate has strong qualities of leadership?

Hillary Rodham Clinton

Donald J. Trump

Joseph R. Biden Jr.

Ben Carson

Jeb Bush

Bernie Sanders

Is your mind made up or is it too early to say for sure?

“I want somebody who is able to stand tall and say there are serious issues in this country and they need to be addressed,” said Ms. Dillard, a retired teacher and computer specialist. “We give Iran billions and our bridges are collapsing? Seriously? I’m more interested in leadership experience than political experience.”

Republicans showing the greatest political momentum in the poll were the three outsiders: Mr. Trump, who drew strong support from self-described moderates, people with family incomes of $50,000 or less, and those without a college degree; Mr. Carson, who edged out Mr. Trump among conservatives and wealthier voters and was much stronger among college graduates; and Carly Fiorina, a former business executive who had support from 4 percent of Republicans, an improvement over her trace amount in August.

“I like Ben Carson because he’s not government, and I’m tired of government,” said Doug May, 70, a retired chief information officer in Bluffton, S.C. Mr. May added, referring to the Russian president, Vladimir V. Putin: “What worries me the most about Donald Trump is what he would do if put into a situation where he is sitting down with Putin. With his mouth, I have no idea what he would say to people.”