WASHINGTON—The Senate this week will grapple with President Donald Trump’s decision to stop making subsidy payments to health insurers, with lawmakers seeking a deal that would keep the money flowing while Republicans try to fold in conservative-oriented health-care priorities.

It remains unclear whether a package could emerge that attracts support from a critical mass of senators and also from House Republicans. That could be put to the test quickly, as Sens. Lamar Alexander (R., Tenn.) and Patty Murray (D., Wash.) are expected...