Mr. Trump pushed hard behind the scenes to ensure that Friday’s symbolic, flag-bedecked signing ceremony with the leaders of Canada and Mexico took place. His staff spent much of the preceding 48 hours coaxing Prime Minister Justin Trudeau of Canada to appear at the event so that three North American leaders, not just Mr. Trump and President Enrique Peña Nieto of Mexico, would appear onstage, according to people familiar with the planning.

But it was strictly ceremonial. Under the United States’ fast-track trade rules, the deal must pass the House, where all bills that affect federal revenue must originate, before moving to the Senate for final passage.

Technically, the deal that the three trading partners signed cannot be altered. But in reality, the administration is free to renegotiate with Canada and Mexico to make changes that could ultimately be included in final legislation. If nothing is passed, Mr. Trump could pull out of Nafta, as he has long threatened, or simply continue operating under the old rules.

The Democratic takeover of the House has created anxiety among business groups and legislators, who are urging Republican leaders to push for a vote on the proposal in the lame-duck session, before Democrats assume control.

“Manufacturers need certainty now, not later,” said Jay Timmons, president and chief executive of the National Association of Manufacturers. “With two million American jobs dependent on exports to Canada and Mexico, Congress needs to move expeditiously.”

Mr. Timmons’s group, along with other trade associations and Republican Senators Lamar Alexander of Tennessee and Patrick J. Toomey of Pennsylvania, are pressing House and Senate leaders to quickly pass the deal.

Senator Sherrod Brown, a Democrat from Ohio, has been quietly pushing a revision to the trade agreement aimed at improving wages and working conditions in Mexico. The provision would require that Mexican imports to the United States be certified as having been produced in accordance with labor standards, according to American officials.