The Daily Mail says Akinfeev will be offered a new contract by CSKA Moscow in a bid to ensure they don't lose him for nothing.
The Russia international is seen as a successor to Edwin van der Sar and his current deal with CSKA runs out in December next year.
Van der Sar's decision to pen another one-year deal at Old Trafford means the Barclays Premier League champions can now bide their time when replacing the 39-year-old.
But CSKA are determined not to let him leave cheaply and are expected to offer an extension through to 2014.
United boss Sir Alex Ferguson was full of praise for Akinfeev earlier this season.