BRUSSELS—The European Union thought it was on the verge of a deal with the U.S. to keep tariffs in check. Now it is readying a counterattack.

U.S. officials told the EU’s trade chief this past week that they expect President Trump to demand further concessions from the bloc to get an agreement, including a baseline tariff on most European goods that could be in the range of 15% or higher, according to people briefed on the talks.