The Quinnipiac University poll findings were similar, with 55 percent Americans believing Trump was not capable to serve as president, while 41 percent said he was.
The CNN poll released on Monday showed that only 32 percent of respondents believed that Trump was "honest and trustworthy".
Earlier this summer, Quinnipiac found 57 percent thought the president was intelligent.
Among independents the drop was sharper, from 47 percent approval last month to 31 percent now.
Eight polling bodies - ABC News/Washington Post, CNN, Gallup, IBD/TIPP, the Kaiser Family Foundation, Quinnipiac University, Grinnell College/Selzer & Co, and Suffolk University - gave the president an average approval rating of 38 percent, a drop of three points from Trump's previous approval polling average, the Vox website reported Tuesday. But Trump got 49 percent approval for his handling of the economy. For example, in one instance claims Woodward, Trump had to be talked out of sending a tweet about North Korea that the regime could have taken as a threat, potentially leading to nuclear war.
This, together with Trump choosing Brett Kavanaugh, an unpopular candidate, as Supreme Court nominee, might make it hard for the administration to project a confident image, which might be reflected in the November polls, Vox reported. Republicans were the only group that trusted Trump more than the media, by 72 to 12 per cent. Holding off electoral disaster becomes even more hard when that president has a low approval rating.
The Quinnipiac poll was conducted between September 6 and 9, smack in the heart of the controversy surrounding The Times op-ed.
"American voters don't want Congress to tell President Donald Trump, 'You're fired!' by serving him with articles of impeachment", Xinhua quoted Tim Malloy, Assistant Director of the poll, as saying.
The White House did not immediately respond to a request to comment on the poll.