Trump remains the GOP front-runner, holding steady in the poll, released Sunday, with the same 27 percent support he saw last month. Carson, a retired neurosurgeon, is still in second place with 21 percent -- he had 23 percent in September.
The next closest competitor is Texas Sen. Ted Cruz, who is polling at just 9 percent -- although that's up four percentage points. Florida Sen. Marco Rubio has 8 percent support, while former Florida Gov. Jeb Bush and former Hewlett-Packard Chief Executive Officer Carly Fiorina each have 6 percent.
New Jersey Gov. Chris Christie has 3 percent in the poll, up from 1 percent in September. Former New York Gov. George Pataki has less than 1 percent.
On the Democratic side, there has been little movement over the past month. Front-runner Hillary Rodham Clinton has 46 percent and Vermont Sen. Bernie Sanders has 27 percent -- matching their September numbers in the poll.
Vice President Joe Biden, who has not announced whether he will run or not, has 16 percent support, up one percentage point from last month.
If Biden decides not to run, Clinton's lead over Sanders would grow from 19 percentage points to 24, the poll found.
As CBS News' Brian Webb reported, the latest CBS News polling also showed voters have trust issues with both Clinton and Trump. Clinton scored just 35 percent among Democratic primary voters for being honest and trustworthy, while on the Republican side, Trump comes in even lower at 33 percent.
Trump remained as optimistic as ever Sunday.
"I happen to be an honorable guy. But one thing that's very important -- when I go to New Hampshire and Iowa and South Carolina, where I have very good numbers in terms of favorability, you know why? Because I'm there a lot," Trump said.
Carson got the best numbers for being honest and caring. He told CBS News' John Dickerson Sunday why he believes Congress is so dysfunctional.
"Well you know, over the last few elections, a lot of people have been sent to Washington with the thought that maybe some changes could be made, and I don't think anyone's seeing any changes," Carson said.
And if Biden runs, he'll have a head start on the honesty issue among Democrats – finishing ahead of all the other candidates at 61 percent.
The first Democratic presidential debate takes place Tuesday night in Las Vegas.
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