"The player will undergo a small surgical cardiological intervention in the coming days and the recovery time will be better defined after the surgery, but it will probably be a few months," Milan said in a statement said.
The problem caused insufficient blood flow to the brain, causing "ischemic cerebral damage" but no lasting brain damage has been detected.
The former Real Madrid striker, who recently said that football had left him exhausted and he would quit in 2014, fell ill after returning by air from Rome after Saturday's 3-2 win over AS Roma.
His team-mates unveiled a shirt bearing his name after scoring in Tuesday's 1-1 draw at BATE Borisov which helped take Milan into the Champions League.
"Antonio is in a good condition, the lad is fine but I'm not a doctor," Galliani told reporters.
"I was very, very worried. His career is not at risk. The doctors say he will need a few months, not too many, although I don't want to say too much at this stage, maybe in four, five or six months he will be in conditions to play.
"He has told me that he's upset because he had been playing well."
Abete said: "The most important thing is his complete and quick recovery, and whenever it will be, it will be," said Abete.
"We will stay in constant contact with the club and will do everything in his interest. His health comes first, the rest afterwards."
The 29-year-old became a key factor in their Serie A triumph in May and has also emerged as Italy's main forward under Cesare Prandelli as they easily qualified for Euro 2012.
Prandelli will now hope the skilful forward will recover in time for next June's finals in Ukraine and Poland.
