WASHINGTON—A bloc of congressional Democrats is bucking White House pressure to exclude concerns over currency manipulation from a bill to expedite a Pacific trade pact, widening a rift between President Barack Obama and his base that threatens the trade deal’s passage.

The White House and Republicans have formed an unusual alliance in favor of fast-track legislation to put the proposed 12-nation trade deal to an up-or-down vote without amendments. But as Mr. Obama seeks to build a coalition to pass the trade bill in coming...