Amorim has been confirmed as Milan's new head coach and previously worked with the Uruguayan at Sporting before taking him to Old Trafford.
United signed Ugarte from Paris Saint-Germain in 2024 for a fee of £42m, but the move has failed to deliver the impact expected.
Much was anticipated of the defensive midfielder given his reputation as one of Europe's most aggressive ball-winners, and that reputation only grew once Amorim's own arrival at United appeared to confirm a long-term role for his former player.
Instead, the midfielder has started just 30 league games across two seasons, and slipped further down the pecking order once Kobbie Mainoo reclaimed his place in the second half of last season.
Michael Carrick has indicated to Ugarte that he is not part of his long-term plans at the club.
United have already brought in Brazilian midfielder Ederson from Atalanta and are preparing a bid for West Ham's Mateus Fernandes, further squeezing Ugarte's path back into the side.
With his future at Old Trafford effectively over, Ugarte has instructed his agent, Jorge Mendes, to find him a new club.
Mendes has since opened discussions with Milan over a possible deal.
Amorim worked with Ugarte during a successful spell at Sporting before the pair were reunited in Manchester, where expectations were high following the coach's arrival.
Despite that familiarity, Ugarte never managed to justify his transfer fee during his time under Amorim at United, with his appearances becoming increasingly limited as the Portuguese coach turned to other options in midfield.
Milan, who failed to qualify for the Champions League after finishing fifth in Serie A last season, are now exploring whether a reunion in Italy could give the midfielder the fresh start he has been unable to find in the Premier League.
For Ugarte, a move to San Siro would represent a chance to rebuild his reputation away from the scrutiny that has followed his time in England.
United, meanwhile, continue to reshape their midfield options as they look to move on from a player who arrived with considerable promise but never settled into the side.