The Russia goalkeeper's representatives met officials from the Old Trafford club this week with a January move on the agenda.
The Dutch veteran is expected to announce soon that this season will be his last.
New CSKA boss Leonid Slutskiy praised Akinfeev's performance at Old Trafford this week: "This is a rare case when a goalkeeper concedes three goals, but, for me, is the man of the match."
Akinfeev has 37 caps and a Euro 2008 semi-final outing with Russia, as well as three League titles, four Russian Cups and the 2005 UEFA Cup with CSKA.
He is constantly compared to the great Lev Yashin and has won the goalkeeper of the year award named after the Soviet legend four times.