Half of those in the poll taken by the Pew Research Centre, 49%, said they believe the president is trustworthy, while almost as many, 46% said he is not.
Bush was at 62% on this measure in a September 2003 Pew poll and at 56% in a Gallup poll in April.
One of Bush's strong suits throughout his presidency has been the perception by a majority of people that he is honest.
The slide in trust in Bush comes at a time the White House is answering questions about top aide Karl Rove's involvement in the public leak of the identity of a CIA operative, which is illegal.
"If the economy were doing better, the Iraq war wasn't as tenuous and people weren't as uneasy about terrorism, then they might be willing to cut Bush some slack on the Rove issue," said Robert Shapiro, who specializes in public opinion at Columbia University.
"It's all tied back to how the war was justified, so it raises all those issues as well"
Robert Shapiro, Columbia University
"And it's all tied back to how the war was justified, so it raises all those issues as well."
Only half the public is closely following the allegations that Rove leaked the identity of a CIA operative, according to Pew. Democrats are predictably more inclined to say Rove should resign than Republicans.
But GOP support for Rove is lukewarm. About four in 10 Republicans said Rove shouldn't resign; about the same number said they were not sure.
Andrew Kohut, director of the Pew Research Centre, said an analysis of the survey suggests the Rove controversy is contributing to the president's credibility problem. The belief that Rove has committed a serious offense is having an impact on Bush's ratings on believability, he said.
The controversy over Karl Rove
is affecting Bush's credibility
Only a fourth of people in an ABC News poll out this week said they believe the White House has been cooperating fully with the investigation of the CIA leak.
About half, 49%, in the Pew poll said they approve of the job being done by Bush on terrorism - as low as he's been on that issue since September 11, 2001. Many independents have abandoned support for Bush in this area.
Bush's job approval in the Pew poll was 44%, with 48% disapproving. The poll of 1502 adults was taken between 13-17 July and has a margin of sampling error of plus or minus 3 percentage points.