Rep. Alexandria Ocasio-Cortez and dozens of her fellow Democrats, who just voted for a bill that falls well short of citizenship rights for illegal immigrants, are now pleading with their Senate colleagues to give them a mulligan and add a path to citizenship back into the legislation.

The House version, which cleared the chamber last week on a 220-213 vote, offers a deportation amnesty and tentative legal status to perhaps 7 million illegal immigrants.

But Ms. Ocasio-Cortez led 90 members who voted for that bill in a letter Monday urging Senate Majority Leader Charles...