The 27-year-old is on the brink of taking a "very big step" in his career by joining the Rossoneri this summer, according to his representative, but CSKA says differently.
The Japan international has been repeatedly linked with a switch to San Siro in recent months, but CEO Adriano Galliani recently threw a proposed deal into some doubt by stating he could wait until the attacker's contract expires in the winter before swooping for the 27-year-old.
However, Honda's brother and representative, Hiroyuki, says the 68-year-old has made a U-turn on past comments by confirming the Rossoneri have lodged a "very good offer" with the Russian giant, which now just has to clear the sale.
"We will meet with CSKA again for more talks, but what I can say is that Milan have a made a very good offer to CSKA," he told Tuttomercatoweb. "I don't know if we can wrap everything up today. We'll look to convince CSKA, because, for us, the deal is ready.
"Joining Milan is not about money, but prestige. Milan, Real Madrid and others are clubs with great value. It would be a big step forward in the career of Keisuke.
"We are waiting for CSKA to give Milan’s offer the OK, then we will be in Italy to finalize everything."
CSKA Moscow sporting director Roman Babaev has offered a conflicting assessment on the deal, though, by claiming his prized asset will not join the Italian giants.
"Honda to Milan? No, this is not true," he told Championat.com.
Honda has netted 19 times in 78 appearances for the capital club since arriving from VVV Venlo in 2009.