The Wales star was informed on Wednesday the reason why the Gunners will not be offering him an extension beyond his contract, which expires at the end of the season.
Ramsey was thought to be demanding a pay packet worth £275,000-a-week but a four-year contract offer was withdrawn in September without an explanation.
Last week, the midfielder appealed for some clarity regarding the decision and has now reportedly been told the club could not afford the terms he wanted.
That revelation will spark a major battle for his signature, which according to The Daily Mail includes Chelsea and United, as well as reigning Serie A champions Juventus and Bundesliga giants Bayern Munich.
The Express understand Liverpool are also interested in acquiring Ramsey, who will be available on a free next summer.
However, teams from overseas are free to begin talks with the 27-year-old from the turn of the year and could get on a jump-start on their English counterparts.
While United missed out on signing Ramsey when he initially joined Arsenal from Cardiff City in 2008, both Chelsea and Liverpool were linked with summer moves for the player.
Former England and Reds midfielder Steve McMahon feels Anfield would be the right destination for Ramsey, who he feels would suit Jurgen Klopp's philosophy.
"Liverpool would do well to get him," he told Sky Sports' The Debate. "He's a midfield player who goes beyond the strikers and Liverpool haven't got that.
"Liverpool have got good midfield players that play in front, they've got top strikers who score goals and have got pace, good defenders.
"But when it's a bit of a stalemate, they haven't got someone to go ahead, the third man running, not just to try and get a goal but to take the players away and open the space up. Ramsey is one of those players and is a goalscoring midfield player with great technique.
"At a club, you've got your levels. Arsenal have their level of a wage structure and Liverpool can well afford the Ramsey's of this world.
"We always look at the high intensity now for Liverpool. I've seen them play in every game this year and they still need a very offensive midfield player and somebody who can take that chance, to get over the top to get away and leave that space for everyone else.
"They haven't got that at this moment in time."
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