WASHINGTON — President Trump’s promise to quickly pass a revamped North American Free Trade Agreement has been upended by the midterm elections, with Democrats who will soon control the House vowing to withhold their support to extract greater protections for American workers.

Administration officials remain confident they will corral the votes for the new United States-Mexico-Canada Agreement, which Mr. Trump speedily negotiated in September to claim a big win on one of his signature issues before the November elections.

While White House officials considered pushing the revised deal through the coming lame-duck session, they did not want to risk a backlash from lawmakers in both parties.

Democrats, emboldened by their midterm win and eager to outshine Mr. Trump as defenders of the American worker, are unlikely to sign off on any deal that does not include significant changes that labor leaders and newly elected progressives are demanding. That could involve reopening negotiations with Mexico, although American and Mexican negotiators have both publicly ruled out that possibility.