Gov. Robert F. McDonnell’s job-approval rating remains near an all-time low, and Virginians take an even dimmer view of his honesty and integrity, according to a poll released Thursday.

Amid federal and state investigations into luxury gifts and loans that a businessman provided to the Republican and his family, a Quinnipiac University poll found that 42 percent of voters said McDonnell was honest and trustworthy, and 41 percent said he was not.

His job-approval rating was higher than that, though little changed from the all-time low he hit last month. In the new survey, 47 percent said McDonnell was doing a good job, and 39 said he was not. In a Quinnipiac poll in July, 46 percent approved and 37 percent disapproved.

Thirty-four percent of Virginians have a favorable view of the governor, and 35 percent do not. As recently as May, he enjoyed nearly 2-to-1 favorability numbers, with 40 percent saying then that they had a positive view of him and 23 percent saying they did not.

“Virginia voters are not giving Governor Bob McDonnell a ringing vote of confidence for his honesty and integrity,” said Peter A. Brown, assistant director of the Quinnipiac University Polling Institute. “His approval rating and favorable/unfavorable ratio are a far cry from when the governor was admired for his job performance and liked by voters.”

Brown added: “The fact that he gets a thumbs up for his job performance but a split on favorability indicates that, to some degree, voters seem able to separate their views of him personally and their evaluation of his ability to run the state. But as the questions about his and his family’s relationship with a prominent campaign donor continue to swirl, he will have a difficult time pushing his numbers up.”

McDonnell’s public image has taken a hit since March, when The Washington Post first reported that he and first lady Maureen McDonnell had promoted a nutritional supplement made by Star Scientific around the time that Star chief executive Jonnie R. Williams Sr. picked up the $15,000 catering tab at the wedding of one of their daughters.

Since then, The Post has reported that Williams gave a $10,000 engagement gift to another daughter, treated Maureen McDonnell to a $15,000 Bergdorf Goodman shopping spree, provided a $6,500 Rolex watch for the governor and supplied $120,000 in loans.

McDonnell has said that Williams received no state favors in exchange for his largess and that any efforts to promote Star were in line with what he and the first lady would do to boost any Virginia-based enterprise.

Yet McDonnell also apologized for embarrassing the state and returned the gifts and money.

Quinnipiac conducted the poll between Aug. 14 and Aug. 19, surveying 1,374 registered voters via land lines and cellphones. The poll has a margin of error of plus or minus 2.6 percentage points.