Virginians headed to the polls Tuesday to elect a new governor in a tight race that is being closely watched as a gauge of the national political mood.

Polls show former Democratic Gov. Terry McAuliffe and Republican Glenn Youngkin, a former private-equity executive, are running neck-and-neck to succeed Democratic Gov. Ralph Northam. Virginia governors are barred from serving consecutive four-year terms.

No Republican has won a statewide race in Virginia since 2009, and the election will test Democrats’ staying power after years of electoral gains. Republicans, meanwhile, hope to reverse losses in the vote-rich suburbs during former President Donald Trump’s tenure.

President Biden, a Democrat, won Virginia by 10 percentage points last year, but his approval ratings have dipped to the low-40% range, making him a potential drag on Mr. McAuliffe. In addition, the state’s off-year gubernatorial contest has historically favored the opposition party of the sitting president, with Mr. McAuliffe’s 2013 victory being the only recent exception.

Mr. McAuliffe, 64 years old, has consistently sought to tie Mr. Youngkin to Mr. Trump, a fellow Republican. Mr. Youngkin, 54, a former Carlyle Group co-chief executive and political newcomer, accepted Mr. Trump’s endorsement but hasn’t campaigned with him.

The former president praised Mr. Youngkin in a telephone rally late Monday, the Associated Press reported. “Tomorrow I’d like to ask everyone to get out and vote for Glenn Youngkin,” Mr. Trump told an invited group of supporters by phone. “He’s a fantastic guy.”