The 41-year-old Amorim replaces Massimiliano Allegri, who was sacked after Milan finished fifth.
Italian media said Amorim, 41, had signed a two-year contract.
"I know exactly what this club means: history, prestige and an extraordinary fanbase around the world. It is a challenge I embrace with pride and enthusiasm, fully aware of what these colours represent," Amorim said.
Milan concluded an abysmal second half of the season with a 2-1 home defeat by Cagliari on the last matchday, winning just one of their final four matches.
The final whistle was met with heavy booing from the San Siro fans as they missed out on a place in Europe's top competition for the second season in a row.
Amorim had a difficult 14-month spell as head coach at United where he struggled to turn their fortunes around.
In October last year, United co-owner Jim Ratcliffe described Amorim as an "outstanding young manager" who needed three more years to prove himself, but over time that faith slipped away and the Portuguese was sacked in January.
Amorim's stock remains high on the back of his achievements at Sporting, where the former Portugal midfielder guided the club to their first Primeira Liga title in 19 years in 2021 before capturing a second league crown in 2024.
Amorim is Milan's fourth manager since the club lifted the Serie A trophy under Stefano Pioli in 2022.
