The 19-year-old moves to United after the two clubs agreed a compensation package, with Swansea entitled to remuneration for his development having been with the club since under-15 level.
Margetson, a Wales under-21 international and the son of former England, Swansea and current Wales goalkeeping coach Martyn Margetson, spent last season on loan at Cymru Premier side Connah's Quay Nomads, helping the club qualify for European competition.
"Manchester United have agreed the signing of exciting young goalkeeper Kit Margetson, subject to registration," read United's official statement.
"The 19-year-old was most recently at Swansea City, a club he'd been with since under-15s. Last season, he had his first taste of senior football on loan at Connah's Quay Nomads and kept 10 clean sheets in 34 appearances, helping the Welsh side to qualify for European competition.
"Margetson has also earned youth international caps for Wales and made his Under-21s debut against Belarus in March. Everyone at United would like to welcome Kit to the club and wish him the very best of luck for the future."
Swansea's own statement confirmed the compensation arrangement and paid tribute to Margetson's development at the club.
"Kit Margetson has joined Manchester United subject to the relevant approval, after the two clubs agreed a compensation package. The Wales Under-21 international has been with the Swans since under-15 level, with the club entitled to remuneration for the player's development.
"He links up with the Premier League Red Devils having spent last season on loan at Connah's Quay, playing an important part in helping them qualify for Europe. Margetson, son of Swansea's former head of goalkeeping Martyn, was included in the first-team matchday squad for a fixture against Millwall towards the end of the 2024-25 season, and was also a regular part of the wider senior squad. Everyone at Swansea City wishes Kit every success during his time at Old Trafford."
Margetson is the second Welsh goalkeeper to join United this summer, following the arrival of Karl Darlow on a free transfer after his exit from Leeds. He is expected to feature initially for United's under-21 side behind senior options Senne Lammens, Darlow and Tom Heaton.