Far more than other Republican contenders, Mr. Carson, a retired neurosurgeon, has capitalized on his outsider message — a mix of anti-establishment views, delivered in a calmer tone than Mr. Trump’s, and socially conservative positions — to draw voters away from rivals and leap ahead in the poll, which will be published on Tuesday. The proportion of Republican voters favoring Mr. Carson rose to 23 percent from 6 percent in the previous CBS News poll, which was taken just before the first televised Republican debate in early August. Over that same period, Mr. Trump made modest gains, to 27 percent from 24 percent.

Mr. Carson pulled at least some of his support from Republicans who are more typical political figures. Jeb Bush fell in the poll, to 6 percent, from 13 percent, and Gov. Scott Walker of Wisconsin tumbled to 2 percent from 10 percent. No other candidates fell as much as those two, according to the poll. Mr. Carson drew more support from college graduates than Mr. Trump, while those without a college education were more likely to favor Mr. Trump.

Over all, 37 percent of Republican voters say their minds are made up about which candidate they will support as their party’s presidential nominee, while 63 percent say it is still too early to say.