The deal was finalised following swift negotiations, with Manchester United keen to strengthen their goalkeeping department ahead of the new season.
Lammens, who has already impressed in the Belgian Pro League with his composure and shot-stopping ability, is regarded as a long-term investment for the club.
United's recruitment team had been monitoring the Belgian for several months, and the agreement with Antwerp includes additional performance-related clauses that could see the final fee rise further.
Lammens will provide extra depth and competition in Ruben Amorim's squad, with his arrival seen as part of a wider strategy to build for the future.
The young keeper, who has represented Belgium at youth level, will now begin his new chapter at Old Trafford as he looks to adapt to the Premier League. United believe his potential and development curve could make him a key figure in the coming years.