Ramsey is set to quit the Emirates next summer, after the Gunners withdrew a contract offer that had been on the table.
The move has put Jose Mourinho and Ed Woodward on high alert, after two failed attempts already to land the midfielder.
Former United boss, Sir Alex Ferguson, wanted Ramsey before he left Cardiff for Arsenal in a £4.8million deal in 2008.
And the UK Mirror reports United tried to land Ramsey last summer, after making a discreet enquiry about the Welshman.
Arsenal are ready to listen to offers for Ramsey in January, but the player says he intends to stay for the rest of the season and achieve “something special”.
Ramsey can sign a pre-contract agreement with a foreign club from January and it’s believed he’s tempted by a move abroad.
United face competition from Liverpool, with Jurgen Klopp also a big fan of the 27-year-old.