WASHINGTON—Senate Republicans’ marathon session stretched into early Tuesday, as party leaders struggled to corral enough votes to pass the party’s “big, beautiful bill” and move the legislation one step further toward President Trump’s desk ahead of lawmakers’ self-imposed July 4 deadline.

Voting on amendments and procedural motions started midmorning Monday and dragged on for more than 24 hours as Republican leaders worked to find a balance of policies that could pass both the Senate and later the House. Trump, who has been speaking with congressional leaders by phone, said Tuesday morning he was confident that the bill was on track even as its fate and final content remained uncertain.