The 28-year-old Wales worldwide had already been told by new Arsenal manager Unai Emery that he did not feature in his plans at the Emirates and will soon bring to an end his 10-year association with the north London club.
Ramsey has always been associated with a move to the Old Lady after his negotiations for a new contract broke down with current club Arsenal.
Other reports suggest that Ramsey will head to Italy on Sunday to complete his medical ahead of the move.
"Aaron Ramsey to on next June, done deal", Italian journalist Fabrizioi Romano tweeted on Wednesday night.
Ramsey joined Arsenal for £4.8million from Cardiff City and has played 252 games for Arsenal, scoring 52 goals.
Ramsey has been linked with several clubs from across Europe and it has been recently reported that Juventus are closing in on a deal to sign him in the January transfer window.More news: Cristiano Ronaldo Denies Threatening to Have a Woman's Body ''Cut Up''
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Despite making clear his future lies away from the Emirates, Ramsey has continued to play a big part in Unai Emery's first season in charge of Arsenal, scoring three goals in 25 appearances.
"My idea and focus, I say to him every day, is with us", said Emery.
"His future is individually for him, for his family, his agent".
"Every training day with us he is working very well and I want his behaviour and focus the same for West Ham on Saturday. We can only loan players", said Emery.
"But between him and us, I am very happy with him now". Let us know in the comments...