WASHINGTON—The White House sent a warning shot to congressional Republicans that it may increase its outreach to Democrats if it can’t get the support of hard-line conservatives, a potential shift in legislative strategy that could affect drug prices, the future of a tax overhaul and budgetary priorities.

Days after the House GOP health bill collapsed due to a lack of support from Republicans, White House Chief of Staff Reince Priebus brought up the idea of working with Democrats multiple times, leaving little doubt that the...