The commission added, “On the contrary, it would immediately result in retaliatory measures by the U.S., leading to the visa requirement being imposed on all E.U. citizens.”

Dimitris Avramopoulos, the commissioner for migration, home affairs and citizenship, said that inflaming the dispute would not help because the union was making progress on visa issues through diplomacy. He noted that Canada had agreed to stop requiring Bulgarian and Romanian visitors to obtain visas as of Dec. 1.

“Our continued engagement and patient diplomatic contacts over the past year have brought tangible results already with Canada, and we are committed to proceeding in the same way with the U.S.,” Mr. Avramopoulos said in a statement. He added, “We are on the right track.”

The commission said it would report to the Parliament again on the matter by the end of the year.

European impatience with the American visa policy has been building for years. The commission considered proposals a year ago to end visa-free entry for Americans and Canadians, but it concluded then that doing so would “have significant negative impacts in a wide range of policy areas, notably on external relations, trade, tourism and the E.U.’s economy.”

The commission said on Tuesday that in recent months, “political and technical contacts have been stepped up” with the new American administration. Mr. Avramopoulos said he had discussed the matter with the Homeland Security secretary, John F. Kelly, in Washington on Feb. 8. The subject also came up when American and European officials met in Malta in March.