United have been long-term admirers of of the Welsh international and twice failed with attempts to sign him in the past.
But news that Arsenal have withdrawn their offer of a new deal for the 27-year-old has suddenly thrown up the chance to snap him up.
And the Mirror say Jose Mourinho and the Red Devils board are considering an attempt to take him on a free transfer at the end of the season.
Ramsey was left stunned last month when talks over a new deal at the Emirates suddenly hit the buffers when the club decided to take their offer off the table.
He had been hoping to agree a lucrative long-term deal to commit the remainder of the top years of his career to the North London side his joined in 2008.
But new manager Unai Emery has decided to let Ramsey run down the final 12-months of his contract and allow him to leave next summer.
Italian giants Juventus had shown an interest in trying to agree a cut-price deal for the midfielder in January.
But Arsenal seem happy enough to allow him to leave without a fee – and last week the player revealed he intends to stay at the Emirates until next summer.
Having proven himself at the top level both in the club game and on the international stage, there is likely to be no shortage of offers from across Europe.
But it is United who could be at the front of the queue for a player they believe would add something to their sometimes laboured midfield.
Should they follow through on their interest it would be the third time they have made a play to sign him.
Sir Alex Ferguson had even agreed a £4.8million fee with Cardiff ten years ago after he had burst onto the scene as a highly-promising teenager.
But at the stage Ramsey opted to join Arsenal instead, believing he would be able to develop better as a player under the guidance of Arsene Wenger.
It is claimed United then made a second attempt to snap him up as recently as last summer when they knew Ramsey was entering the final year of his deal.
But the Gunners rejected the advances, even though it means they have written off around £25m they could have got for the player rather than let him leave without a fee next year.
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