Ramsey looks set to leave Arsenal either in January or next summer after failing to agree terms on a new contract.
Unai Emery is thought to be the reason behind Arsenals reluctance to give in to Ramseys contract demands.
Ramsey wants £185,000-a-week to stay at Arsenal, but Emery is keen to spend the funds available on new signigns.
The Mirror claim United will make a third bid to bring Ramsey to Old Trafford.
United failed in 2008 to sign Ramsey ahead of Arsenal and saw an approach for the Welshman knocked back last summer.
Liverpool too were thought to make enquiries about Ramsey in the summer, but Arsenal rejected those moves as Emery wanted to assess the squad he inherited from Arsene Wenger.
Emery and Ramsey did not immediately click and reportedly had a falling out when the new Arsenal boss dropped the midfielder for a clash with Chelsea in August.
Ramsey is free to speak to foreign clubs in January as his contract expires next summer.
Arsenal may be tempted to cash in on Ramsey, rather than lose the 27-year-old for nothing at the end of his contract.
While transfer speculation continues to mount, Ramsey insisted this week he has no plans to leave Arsenal in January.
Will I stay for the rest of the season? Of course, yeah, he said on international duty.
I am contracted to Arsenal and I am going to do my best now this season to try and achieve something special.