Arsenal and Wales midfielder Ramsey, 25, is taking the Euros by storm and Mourinho wants him to be Manchester United’s answer to Frank Lampard.
Mkhitaryan, 27, remains his top summer target, but Dortmund are playing hardball over a fee for their attacker.
United have tabled an improved bid of £28m for the Armenia captain, after an initial offer of £19.3m was rejected.
But boss Mourinho fears Dortmund will dig in over a player who last season bagged 19 goals and provided 24 assists.
The German giants have already lost Ilkay Gundogan to Manchester City and Mats Hummels to rivals Bayern Munich.