Oliver Glasner, Andoni Iraola and Alvaro Arbeloa had all been mentioned as possible options for the role in recent weeks.
It was confirmed on Tuesday, however, that Ruben Amorim will take over as head coach, signing a contract that runs until the summer of 2029.
Amorim had been without a club since being sacked by Manchester United in the winter, having previously spent several years in charge of Sporting Lisbon before his move to England.
According to Sky Sport Italia, the 41-year-old wants to bring two players with him to AC Milan that he previously coached during his time in Portugal.
The targets are midfielder Morten Hjulmand, 26, and attacking winger Francisco Trincao, also 26.
Reports suggest both Hjulmand and Trincao are among Amorim's leading priorities for new signings this summer as he begins reshaping the Milan squad.
Trincao has recently also been linked with several leading English clubs, including Liverpool, Chelsea and Arsenal, adding competition to any move.
The Portuguese winger's contract with Sporting runs until the summer of 2030, giving the Primeira Liga club a strong negotiating position over any sale.
Hjulmand has long been linked with a departure from Sporting, with reported interest in the Danish midfielder previously coming from clubs including Liverpool, Manchester United and Atletico Madrid.
His contract with the Portuguese club runs until the summer of 2028.
Amorim's familiarity with both players from his time at Sporting is understood to be a significant factor behind his interest, with the new Milan coach keen to build his squad around players whose qualities and characteristics he already knows well.
Securing either player is likely to require Milan to match Sporting's valuation given the level of interest already shown by other clubs across Europe, though Amorim's personal relationship with both players could prove decisive in any negotiation.
AC Milan's hierarchy are expected to back the new coach's transfer requests as he looks to make an immediate impact following Allegri's departure and the disappointment of last season.
Hjulmand has established himself as one of the most complete defensive midfielders in Portuguese football since his move to Sporting, with his physical profile and reading of the game seen as a natural fit for Amorim's tactical demands.
Trincao, meanwhile, has continued to develop his reputation as a creative attacking threat from wide positions, with his end product and dribbling ability making him a sought-after target across multiple leagues this summer.