Marsh joined Swansea from Huddersfield Town in June 2022.
There was speculation about his future after the Welsh club appointed Paul Watson as their sporting director last summer.
Marsh ultimately remained at Swansea working under Watson, but he will now depart.
When he was appointed, Marsh was put in charge of all football departments at Swansea, including player recruitment.
But Watson's arrival from Luton Town saw him assume responsibility for all Swansea's football operations, with what the club described as a "particular focus" on player recruitment.
Swansea, who have Chelsea defender Bashir Humphreys on a year-long loan this season, have been criticised for their transfer work in recent windows.