WASHINGTON—President Donald Trump on Friday upended delicate House negotiations over immigration legislation by suggesting he wouldn’t sign a compromise measure, prompting a clarifying tweet from the president and leaving Republicans uncertain about how to proceed.

In a morning interview with Fox News, Mr. Trump had indicated he wouldn’t sign a “moderate” immigration bill, a remark that was taken by many to mean he was rejecting a recently released bill assembled by negotiators from the GOP’s moderate and conservative wings....