WASHINGTON—The Obama administration agreed to back the lifting of United Nations sanctions on two Iranian state banks blacklisted for financing Iran’s ballistic-missile program on the same day in January that Tehran released four American citizens from prison, according to U.S. officials and congressional staff briefed on the deliberations.

The U.N. sanctions on the two banks weren’t initially to be lifted until 2023, under a landmark nuclear agreement between Iran and world powers that went into effect on Jan. 16.

The...