WASHINGTON — President Obama’s most aggressive and sustained legislative push since the Affordable Care Act faces a crucial first test this week when a divided Senate considers a bill that would grant him accelerated power to complete a massive trade accord with 11 nations across the Pacific Rim.

But after lobbying members of Congress in a campaign that has included rides on Air Force One, meetings in the West Wing, private vows of political support and public attacks on critics in his own party, Mr. Obama’s top legislative priority remains at risk.

A vote scheduled for Tuesday on legislation that would grant him trade promotion authority, also known as “fast track,” has become mired in a procedural thicket, with Democrats — many of them loyal to labor unions bent on killing the bill — vowing to oppose it.

Once Congress grants a president trade promotion authority, lawmakers have the ability to vote up or down on a final trade agreement, but they forfeit the right to amend the deal or filibuster it. The bill before the Senate adds a new twist: If lawmakers decide a final trade accord falls short of their standards, Congress can vote to revoke the president’s authority and then try to amend the deal.