The Red Devils efforts for Griezmann in the summer were stalled when Atletico Madrid were handed a transfer ban which would have made it impossible for them to find a suitable replacement.
But it is believed that now Griezmann will be allowed to switch clubs with the arrival of Costa in January, .
Chelsea confirmed last week that the deal for the Spanish striker is almost complete barring Costa agreeing personal terms and undergoing medical in Madrid.
Griezmann is likely to cost a hefty £90 million as he signed an extension to his Atletico Madid contract in June.
But rumours suggest Mourinho is going to welcome the opportunity to bolster his front-line further.