WASHINGTON—President Trump’s push to revamp North America’s trade rules is hitting a roadblock in Washington as Democrats and labor groups demand changes, dimming its chances of passage before next year’s presidential election.

As Congress returns from recess this week with a full plate of priorities, House Speaker Nancy Pelosi (D., Calif.) and other prominent Democrats have signaled they won’t allow a vote on the administration’s new agreement with Canada and Mexico without certain changes.

Democrats...