Because they have close ties to the unions, Democrats are concerned that additional closures like those in Chicago could lead to a possible replay of the party’s recent loss in Virginia’s governor race. Polling showed that school disruptions were an important issue for swing voters who broke Republican — particularly suburban white women.

“It’s a big deal in most state polling we do,” said Brian Stryker, a partner at the polling firm ALG Research whose work in Virginia indicated that school closures hurt Democrats.

“Anyone who thinks this is a political problem that stops at the Chicago city line is kidding themselves,” added Mr. Stryker, whose firm polled for President Biden’s 2020 campaign. “This is going to resonate all across Illinois, across the country.”