The Sunday Express says Ramsey is in the last year of his Gunners contract and they are ready to sell to finance their own transfer ambitions.
Chelsea boss Maurizio Sarri has accepted defeat in his interest in Aston Villa's Jack Grealish, who will head to Tottenham – if he can persuade Villa to sell him – so he can play in the Champions League.
Sarri, who has missed out on several Italian targets, will turn to Ramsey, who is yet to sign a new deal at Arsenal.
Arsenal's locker room are urging Ramsey to sign a new deal.
Henrikh Mkhitaryan recently stated: "I think he's been with the club for 10 years already and it seems that he became a symbol for this club.