STV reported that only the finer details of the deal are still being discussed, with the former St Mirren player expected to undergo a medical.
Subject to receiving a visa, he will travel to Miami to complete the move to the club owned by David Beckham.
Morgan has been unable to make a sustained run in the starting line-up since signing for Celtic in January 2018. He was initially loaned back to the Paisley club, while another loan spell at Sunderland followed.
He has made 18 appearances for Celtic this season under Neil Lennon, starting the League Cup final against Rangers, but he hasn't played a domestic game since.
Inter Miami are the newest MLS club and will make their league debut in the 2020 season.