These women are beginning to offer themselves as potential presidents at a time when stark divides around gender are shaping the midterm campaigns: A record number of women are running for Congress, mainly on the Democratic side, and polls show women favoring Democrats by a huge margin. Yet Mr. Trump has begun sharply assailing the #MeToo movement and making increasingly explicit appeals to male identity…

If multiple women run, no one Democrat could monopolize the vision of breaking a glass ceiling, as Hillary Clinton did in 2016. And any Democratic woman might face anxiety, expressed quietly by some concerned party members, about the ferocity with which Mr. Trump has savaged his female critics.

But some Democrats say electing a woman is even more important now than in 2016. And many Democratic leaders believe the political mood in the party could quickly catapult one or more women to front-runner status.