The Spain international is set to join the Serie A club on a two-year loan deal this week after negotiating an exit from Chelsea, three years after joining the club in a then British record £50 million deal from Liverpool.
And Galliani has expressed his delight at finally getting his man, revealing that the club wanted to buy Torres from Atletico Madrid in 2007.
"We are very satisfied," Galliani told reporters.
"He was already very close to joining in Milan in 2007 but he was too expensive as he would have cost €37m (£29m). Now we’ve got him for free."
Torres arrived in Milan on Saturday afternoon ahead of a medical that sees him complete his move to Italy, ending a misfiring spell in London that saw him net just 20 goals in the league for Chelsea.
But new coach Pippo Inzaghi claims Chelsea boss Jose Mourinho gave the 30-year-old Spain international a glowing reference.
"I spoke to Mourinho to find out about what kind of man Torres is," Inzaghi told a press conference.
"He is a player who has scored lots of goals and you don’t forget that all of a sudden. He has the right attributes from a human point of view and that is good."