WASHINGTON—Democratic and Republican leaders are weighing their next political steps on abortion after a leaked draft opinion suggested that the Supreme Court was poised to overturn the landmark Roe v. Wade decision.

The prospect of such a ruling just months before midterm elections that will determine which party controls Congress has Senate Democratic leaders planning to put on the floor next week a bill to codify the existing abortion right laid out by Roe and related court decisions. It has no chance of passing because of Republican opposition in a 50-50 Senate, where procedures require a supermajority to advance most legislation. But Democrats could point to the blockade in an attempt to energize voters, particularly as some GOP-led states mobilize to enact laws banning the procedure.