WASHINGTON—Lawmakers again fell short of an agreement on the next round of coronavirus emergency aid Thursday, as a $350 billion loan program for small businesses ran out of money and millions more Americans filed for employment benefits, bringing fresh urgency to the negotiations.

Republicans want to approve $250 billion more for the small-business loans, meant to support worker paychecks, before moving on to other aid and stimulus proposals. Democrats want to make changes to the small-business program and include money...