The Welshman has one year left on his current deal and has not yet committed his future to Arsenal, meaning that the Gunners may be forced to sell if they don't want him to leave on the cheap.
They are hoping to avoid another Alexis Sanchez situation. Arsenal turned down a £55m bid from Manchester City for the Chilean last summer but were then forced to sell him in January to Manchester United in a swap deal with Henrikh Mkhitaryan.
Ramsey is valued at £40m and any fee received would go straight into next season's transfer budget as Arsenal look to rebuild under their new manager, who hasn't been announced yet. The Gunners are reportedly planning a defensive makeover, with three new signings targeted.
The Sun reports that Chelsea, Manchester United and Juventus will all be keeping close tabs on the Ramsey situation and may consider a move for the 27-year-old if Arsenal do indeed decide to sell.
I am honoured to have captain your final game boss. I’m delighted we got the win for you and finally we won away!! All I can say is I will forever be grateful for you giving me the opportunity to play for this club and believing in me through my career. I wish you all the best... pic.twitter.com/4URVzap1HG— Aaron Ramsey (@aaronramsey) May 13, 2018
Ramsey is currently earning £100,000 a week at the Emirates but is demanding an increase in response to the bumper contracts signed by Mkhitaryan, Pierre-Emerick Aubameyang and Mesut Ozil.
Since joining the Gunners from Cardiff for £4.8m in 2008, Ramsey has made over 320 appearances for the club, scoring 58 times including the winning goals in the 2014 and 2017 FA Cup finals.
11 goals made 2017/18 the second most prolific season of Ramsey's career but it wasn't enough to save Wenger's job as he was sacked after Arsenal missed out on Champions League football.
Ramsey's former teammate Mikel Arteta is favourite to take over.