"I urge President Rouhani, as I have done throughout this week, to release Saeed so he can return to our home and our family in the United States."
The ACLJ is a conservative Christian law firm and advocacy group founded by television evangelist Pat Robertson. It has campaigned widely for Abedini's release.
Abedini, a Christian pastor, was sentenced to eight years in prison earlier this year, accused of attempting to undermine the Iranian government.
At the time he was convicted, his wife said Abedini had been arrested by Iranian authorities nearly 10 times before, and had agreed during one detention to stop supporting Christian home churches, and had subsequently taken nine trips to the Islamic republic before he was arrested again in July 2012.
Hekmati, a 29-year-old former U.S. Marine, has been jailed in Iran since 2011 and accused of being a CIA spy. Earlier this month, he said in a letter to U.S. Secretary of State John Kerry that a confession he made nearly two years ago came under duress.
Born in Arizona and raised in Nebraska before settling with his family in Flint, Michigan, Hekmati joined the Marines in 2001 out of high school. He finished his service four years later as a decorated combat veteran with tours in Iraq.
Afterward, he translated Arabic as a contractor and helped train troops in cultural sensitivity.
Hekmati's family said he had gone to Iran to visit his grandmother but was arrested in August 2011, accused by Iran's Intelligence Ministry of working as a CIA agent.
In December 2011, Hekmati appeared on Iranian state television maintaining that he had been sent to Iran by the CIA, but in his letter to Kerry, published on September 11 in the Guardian, he said he had been held "on false charges based solely on confessions obtained by force, threats, miserable prison conditions, and prolonged periods of solitary confinement."
"This is part of a propaganda and hostage taking effort by Iranian intelligence to secure the release of Iranians abroad being held on security-related charges," Hekmati's letter stated.
The Guardian reported that the letter had been smuggled out of jail. Hekmati's family vouched for the letter's authenticity to CNN.