Roma have just one win from their six league matches so far and a growing injury list is not aiding Mourinho's cause, particularly in defence.
No financial details were disclosed, but Italian media reported that the 27-year-old has joined Simone Inzaghi's side for a fee of around 32 million euros ($34.94 million), signing a five-year contract.
The 35-year-old arrives in Milan after an eight-year spell at Juventus, which ended at the end of June when his contract expired.
The Georgia international had an impressive season, helping Napoli secure their first Scudetto in 33 years with his contribution of 12 goals and 10 assists in 33 games.
Mass street partying, punctuated by fireworks, flares, horns and firecrackers, began late on Thursday after the Napoli team secured the title with a 1-1 draw away at Udinese in northeast Italy.
"I'm not stupid, you know," Mourinho said. "Today, I went to the game with a microphone. I recorded everything. From the moment I left the locker room, to the moment I returned. I protected myself."
Runaway leaders Napoli could secure the club's third title and first since 1990 this weekend if they beat Salernitana and second-placed Lazio drop points against Inter Milan at San Siro.
Luciano Spalletti's side extended their lead at the top of the table to 19 points over second-placed Lazio, who will be in action against Juventus on Saturday.
Milan are now third in Serie A with 52 points, 22 points behind runaway leaders Napoli. They trail second-placed Lazio, who face Juventus on Saturday, by three points.
Napoli are still top of the table with 41 points after 16 games but their advantage over second-placed AC Milan, who beat Salernitana 2-1, narrowed to five points.
Of the big five leagues, only the Premier League saw an increase in clubs' cumulative operating profits, up from 49 million pounds to 479 million.
Inter struggled considerably in the first half against an Empoli side who have won just once in their last 20 games.
An AC Milan spokesperson said only that "AC Milan remains focused on improving its performance on the pitch and to develop the club."
AC Milan slipped up for the second time in as many weeks on Friday with a 1-1 draw with Udinese which gave their Serie A title rivals a chance to reach them at the summit.
The champions are on an unstoppable run of league form, winning their last five games without conceding a goal and going ten matches without defeat.
Simone Inzaghi’s side are third with 24 points, four behind leaders Napoli and second-placed AC Milan, who face Salernitana and AS Roma respectively later on Sunday.
Inzaghi takes defending Serie A champions Inter Milan to former club Lazio this weekend.
Allegri's side took down the European champions with a disciplined performance that sucked the life out of the Blues.
The two-time Champions League winner's arrival came like a bolt from the blue for Roma's trophy-starved supporters.
The Middle East and North Africa (MENA) region is the main TV international package still to be awarded by Serie A ahead of the season start.