More than one-third of the respondents said Trump's treatment of the investigation lowered their opinion of him.
Americans also believe that he's been criticized more than recent presidents.
CBS found that the most popular reason people cited for their disapproval was Trump's potential ties to Russian Federation. Most who approve of the President say it is not serious.
A CBS News poll shows the president's approval rating is down to 36 percent, 57 percent of Americans polled disapprove of the job he's doing.
CBS News' report added that over recent presidents have seen their support reach similar levels with their party's voters, which is true.
Obama: Senate bill is a tax cut, not health care
McConnell must navigate a narrow route in which defections by just three of the 52 Republican senators would doom the legislation. Susan Collins, R-Maine, said the centrist has some misgivings about the bills as well.
The June 18, 2010 Rasmussen poll - taken at the same point in Obama's first term - showed the former president with an approval rating of 41 percent compared to a disapproval rating of 58 percent.
The president claimed his critics are not stopping him from getting on with his promise to "Make America Great Again", when he fired up his Twitter account on Sunday morning.
Republicans approve of how Mr. Trump is handling the Russian Federation issue, but in far lower numbers (at 57 percent) than their support for the job he's doing on the issues of the economy (80 percent) and terrorism (77 percent). Another 27% said the investigation is serious but not as serious as other issues.
Thirty-nine percent said the investigation is a critical natural security issue, although 32% said it is a political distraction. Views among whites with a college degree remain more negative. A vast majority of Americans from both parties - 81 percent - think Trump should not try to stop Special Counsel Robert Mueller's investigation, and 56 percent of respondents said they think the investigation will be impartial.
CBS found Trump's approval ratings have been hurt by his reaction to the multiple investigations of Russian interference in last year's presidential election, which he repeatedly has described as a "witch hunt" and an excuse by Democrats to explain former US secretary of state Hillary Clinton's stunning upset loss to him.
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