The Daily Mail says the Germans are lining up a bid of about £15million for the 23-year-old, though CSKA Moscow have been asking in excess of £20m.
United and Arsenal have been linked with Akinfeev for some time though there are doubts in Russia whether he is really up to such a step up.
Akinfeev is considered an excellent shot stopper but is not the tallest keeper and has some problems with crosses.
Bayern have found it hard to replace legend Oliver Kahn with many at the German club unimpressed by Michael Rensing, while 35-year-old back-up Hans-Jorg Butt doesn't have too many more seasons left in him.
Akinfeev said: "CSKA is a great club but every player would love to prove their worth at United."