Christian Abbiati has returned to AC Milan as their new club manager, just a year after the former goalkeeper left the club.
Luciano Spalletti wants to "send the fans' hearts racing" again after being officially presented as Inter Milan's new coach.
Bayern Munich winger Costa played for Lucescu at Shakhtar Donetsk, and the coach described him as "one of the purest talents I have ever coached."
Sweden international Forsberg, 25, played a key role as Leipzig finished second in their debut Bundesliga season.
In adding Andre Silva, AC Milan continue to sign young talent this offseason, but the contract situation of Gianluigi Donnarumma hangs over the club.
Roma hired Sassuolo manager Eusebio Di Francesco on Monday as their new head coach, replacing Luciano Spalletti, who took over Inter.