WASHINGTON—Senior Justice Department officials objected to sending a plane loaded with cash to Tehran at the same time that Iran released four imprisoned Americans, but their objections were overruled by the State Department, according to people familiar with the discussions.

After announcing the release of the Americans in January, President Barack Obama also said the U.S. would pay $1.7 billion to Iran to settle a failed arms deal dating back to 1979. What wasn’t disclosed then was that the first payment would be $400 million...