WASHINGTON — President Trump began his all-hands meeting with Republican senators at the White House on Tuesday by saying they were “very close” to passing a health care bill, just as efforts to fast-track a vote this week collapsed.

If Republicans do manage to broker a deal — as Senator Mitch McConnell of Kentucky, the majority leader, pledged to do during a lively East Room back-and-forth with the president — it is not likely to be because of Mr. Trump’s involvement. Until Tuesday afternoon, the president was largely on the sidelines as the fate of one of his most important campaign pledges played out.

Mr. McConnell, who kept the president at a polite arm’s length while he oversaw negotiations over the bill, asked Mr. Trump to arrange the meeting with all 52 Republican senators during a morning phone call, in part to show senators the White House was in fact fully engaged, according to two people with knowledge of the call.

When asked by reporters clustered on the blacktop outside the West Wing if Mr. Trump had command of the details of the negotiations, Mr. McConnell ignored the question and smiled blandly.