Two prominent House committee chairs are among the GOP lawmakers now backing the legislation. But after meeting with fellow Republican leaders, Johnson said they were instead moving toward a plan that would put the bill up for a vote Tuesday under so-called suspension of the rules, a simpler fast-track procedure requiring a two-thirds majority vote for passage. Senate Majority Leader John Thune isn’t currently expected to weigh in on whether he will bring up the Epstein measure until after it passes the House, according to a person granted anonymity to disclose internal strategy. But Trump’s support for passing the Epstein files.