Mr. McAuliffe, who is in a dead heat with Mr. Youngkin in public and private surveys, is close to House Speaker Nancy Pelosi and a number of White House officials. He and his advisers have been blunt with Biden aides about the closeness of the governor’s race and have argued that the souring political environment for Democrats is the reason that the contest has grown more competitive, according to party officials familiar with the conversations.

With his state’s voters already casting ballots, Mr. McAuliffe is eager for House Democrats to pass the $1.9 trillion infrastructure bill, which cleared the Senate with 69 votes this summer. Ms. Pelosi promised a band of centrist lawmakers last month that she would bring the measure to a vote by Monday. But with progressives vowing to vote down the infrastructure bill until a vote is held on the larger social-welfare legislation, that timing is now up in the air.

“We’re desperate for this,” Mr. McAuliffe said of how he and other current governors view the public works measure, adding: “We need to fix our roads, bridges. This is too important.”

His fellow moderates, if not quite feeling the same level of political urgency, agree and are perplexed by Mr. Biden’s failure to press both Ms. Pelosi and recalcitrant progressives to approve the infrastructure bill and provide him with a substantial, and much-needed, victory.