If the election were today, President Obama would win the critical swing state of Virginia over Republican Mitt Romney, a new poll shows. And that wouldn’t change even if Virginia Gov. Bob McDonnell joined Romney on the ticket.

Obama holds a 47 percent to 42 percent lead over Romney in the Old Dominion, according to a Quinnipiac University poll released Thursday. Obama's lead has narrowed since March, however, when he Romney 50 percent to 42 percent. Romney has since clinched the Republican nomination while Obama has been dealt a blow by the most recent jobs report.

Obama’s standing among women - he holds a 51 to 35 percent advantage over Romney - continues to help him in battlegrounds like Virginia. Men prefer Romney 49 percent to 44 percent.

Virginia is one of several key states on this year’s electoral map. A Romney loss in Virginia would make it difficult to overtake Obama in November.

But if Romney is hoping a home-grown running mate would help capture the commonwealth, he should think again, pollsters said. McDonnell as a vice presidential candidate doesn’t improve Romney’s chances in Virginia, the poll shows.

Quinnipiac University's poll on Wednesday showed McDonnell’s standing among Virginia voters at its lowest point of the past year. McDonnell’s job approval rating sits at about 53 percent after a tumultuous legislative session forced him to take positions on a number of controversial social bills.

Independent voters, a key voting bloc to watch moving forward, like Obama over Romney, 45 percent to 37 percent.

A majority of voters said Obama’s support of same-sex marriage and Romney’s opposition to it won’t affect their vote.