WASHINGTON—An intra-party rebellion is breaking out over Democratic National Committee Chairwoman Debbie Wasserman Schultz’s refusal to sanction more than six presidential debates, amid concerns that Republican candidates are being showcased in events that are drawing millions of television viewers.

Donors, candidates, party activists and DNC officials are pressuring Ms. Wasserman Schultz to relent, and some party leaders say they may try to force her hand at a DNC meeting in December by putting the issue of adding debates...