WASHINGTON—Nearly three weeks after Republican infighting sank an overhaul of the Affordable Care Act, President Donald Trump dug back into the battle on Wednesday, threatening to withhold payments to insurers to force Democrats to the negotiating table.

In an interview in the Oval Office, Mr. Trump said he was still considering what to do about the payments approved by his Democratic predecessor, President Barack Obama, which some Republicans contend are unconstitutional. Their abrupt disappearance could trigger an insurance...