House Speaker Nancy Pelosi is set to unveil House Democrats’ drug-pricing proposal officially in coming weeks but its scope and timing may leave it struggling to gain support.

Pelosi’s plan, a targeted Medicare negotiation bill taking aim at drugs that have been on the market without competition, is slated to go through two key health committees in the House after it’s introduced. It’s also expected to be designated H.R. 3, a symbolic gesture noting it as part of the major bills making up Democrats’ domestic policy agenda.

However, progressives in the House are already grumbling that the proposal doesn’t go...