Spalletti has just led Napoli to the Serie A title for the first time since 1990 but there has been speculation about whether he would stay on.
Napoli's triumph ended their long wait for the Scudetto since Argentine great Diego Maradona led them to two championships in 1986-87 and 1989-90.
Berlusconi, who owned AC Milan for three decades, bought Monza in 2018.
Milan's Niguarda hospital said its trauma team had reconstructed two muscles in the back of the 29-year-old Spanish centre-back.
Dybala will become Roma's fourth signing of the transfer window after the arrivals of midfielder Nemanja Matic, goalkeeper Mile Svilar and defender Zeki Celik.
Olivier Giroud scored a brace as Milan defeated Sassuolo 3-0 on the final day.
The Argentine, who was also a professional footballer and joined Diego at Napoli in 1987 aged 18, died at his home near Naples.
It was the first time three generations of the same family have started in Serie A.
The 23-year-old on Wednesday completed his move from Sassuolo.
Roma confirmed the 40 million euro ($47.08 million) deal for the 23-year-old, who joins Jose Mourinho's team in search of game time.