The Italian champions have been linked with a move for the out-of-favour forward for several months and look to have edged ahead of rivals Inter in the quest to bring the 27-year-old to San Siro.
However, a deal now needs to be reached with the current Premier League leaders, with Galliani confirming Milan are looking to take Tevez on loan with a view to a permanent transfer.
"We have an agreement with the player," Galliani told Sky Sport Italia.
"There are ongoing negotiations with Manchester City and we expect an answer on a loan proposal with the right to purchase the player later on very soon."
The club's chief executive also reaffirmed that Alexandre Pato is staying put in Italy, despite strong interest from Paris Saint-Germain who are now coached by former Milan man Carlo Ancelotti.
"There is no negotiation for the sale of Pato," Galliani said.
Milan return to Serie A action on Sunday when Massimiliano Allegri's men visit Atalanta.
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