Liberal Democrats are intensifying their pressure on Hillary Rodham Clinton to oppose President Obama’s Pacific trade deal as detrimental to American jobs. But Mr. Obama’s allies want her to endorse the accord, which the president has called a boon to the United States economy.

And Mrs. Clinton, stuck between the progressives she must woo in a Democratic nomination fight and the president under whom she served, has remained, for the most part, mum.

The issue has become the first major policy test in her fledgling campaign, with Mrs. Clinton under mounting pressure to pick a side in the delicate and heated debate over the Trans-Pacific Partnership deal, a 12-nation trade agreement that Mr. Obama has aggressively pursued and that faced a critical vote in Congress on Tuesday.

Just 48 hours after Mrs. Clinton delighted liberal Democrats with a proposal to expand citizenship eligibility to immigrants who are in the country illegally, protesters on Thursday urged her to speak out against the trade deal.