Ramsey appears to be on his way out of Arsenal in the summer and it would be remarkable if the Merseyside club could acquire his signature.
Arsenal have a five-point lead over Manchester United in their pursuit of a top-four finish to secure a return to the Champions League after a two-season absence.
One midfielder that looks to be on his way out of the club is Aaron Ramsey, with discussions to extend his current contract now at a standstill.
Ramsey has already insisted he has no plans to leave Arsenal in January, but did admit he thought he had agreed a new deal before the Gunners called a halt to talks.
Liverpool and United are now looking to take advantage of this by making a cut-price move for the Welsh worldwide - offering Arsenal a transfer fee instead of them letting him leave on a free.More news: Chiefs come up short against Patriots
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With the likes of Juventus also keen to sign the player, Arsenal fans will be hoping that the midfielder turns down Premier League interest to move overseas, but that may not be the case.
The Mirror claim that Manchester United are ready to make a fresh attempt to sign Ramsey after missing out on the player to Arsenal ten years ago but they'll face stiff competition from rivals Liverpool.
However, the report says Mourinho will urge Man Utd chiefs to do all they can to land Ramsey ahead of Liverpool as the Portuguese coach is a long-term admirer and feels the Welshman would be a great addition to his squad.
Ramsey could also negotiate with a foreign team from January, with Juventus reportedly interested.
Talks between Arsenal and Ramsey broke down last month.