WASHINGTON — Republican leaders are planning a House vote next week on a bill to ban nearly all late-term abortions after dropping a requirement for reporting rapes that sparked a January rebellion among Republican women and moderates.

After working behind the scenes with anti-abortion groups and female Republican lawmakers, party leaders have readied legislation allowing rape victims to have late-term abortions if they receive counseling or medical treatment at least 48 hours before the procedure. That replaces language that only allowed abortions for rape victims who reported the assault to law enforcement authorities.

Republican women and moderates had objected that requiring a report to law enforcement officials placed an unfair burden on women already staggered by the extraordinary stress of a sexual assault and resulting pregnancy. Some also said the earlier provision could make the Republican Party look harsh as it seeks to win support from women and younger voters for the 2016 presidential and congressional elections.

The overhaul of the bill highlighted the competing political pressures that Republicans confront. They also remain eager to avoid alienating anti-abortion voters who are among the Republican Party’s most staunch backers.

The bill would also let minors who are victims of incest have abortions if they report the attack to social service workers or law enforcement agencies. The measure was described by Republican aides and lobbyists who spoke on condition of anonymity because its details were not publicly released.

Republican aides predicted House passage of the legislation, which would ban most abortions starting with the 20th week of pregnancy. The vote could occur on Wednesday, the second anniversary of the murder conviction of a Pennsylvania abortion doctor, Kermit Gosnell, in the deaths of three babies who were delivered alive and later killed with scissors.

Anti-abortion groups and lawmakers who worked with Republican leaders praised the legislation.

Carol Tobias, president of National Right to Life, called the measure “a strong bill.”

Democrats and abortion-rights groups opposed the initial bill and are likely to do the same now.