President Donald Trump said Tuesday he was optimistic that the Senate would vote to begin debate on legislation that would do away with the health-care law enacted by his predecessor, a central promise of his campaign.

Mr. Trump, in a 45-minute interview with The Wall Street Journal, said he has been working to secure the 50 votes needed to pass the measure, which he described as a difficult but crucial step toward ultimately putting in place a health-care system that would replace former President Barack Obama’s Affordable...