BRUSSELS—U.S. and European trade negotiators face growing domestic pressure over agriculture, with clashing demands threatening to rekindle a tit-for-tat economic war.

The food fight comes as U.S. farm lobbies, Congress and some Trump administration officials demand access to European markets following a trans-Atlantic trade truce in July. European Union officials are refusing to engage, arguing the White House agreement specifically omitted agriculture and focused instead on slashing tariffs on industrial goods.

U.S....