Democrats, planning to put a series of health-care bills onto the House floor in coming weeks, aim to pressure Republicans by bundling some drug-pricing measures that conservatives support with moves to shore up Obamacare.

That would force some Republicans into the tough decision of opposing measures to reduce prescription drug prices, which both parties have promised to do, to avoid supporting a bill bolstering Obamacare’s individual health insurance markets, according to a senior Democratic aide. It’s also partly a practical...