The Wales international has little over 12 months to run on his current deal at the north London side ahead of the new season.
Ramsey could leave for nothing next summer unless the 27-year-old agrees to a new deal with the League Cup runners up.
The former Cardiff City star has been linked with a potential switch to Chelsea FC and Liverpool FC in the summer.
Arsenal appointed Unai Emery as Arsene Wenger’s replacement at the end of the 2017-18 season following a poor campaign.
Sunday Times reporter Castles believes Arsenal will sell Ramsey if the Welsh star won’t commit to the north London side.
“It is a big warning signing when Ramsey is talking about his agent having discussions with Arsenal on the final days of the transfer window,” Castles told the Transfer Window podcast.
“It suggests to me the club is worried and could cash in. They realise he could leave and there is a danger of losing him for nothing.
“They want to find out if they can tie him down now. If they can’t, they would look to see him now.”
Arsenal finished in sixth place in the Premier League table last season to miss out on a Champions League spot.
The Gunners lost 2-1 to Atletico Madrid in the Eurppa League semi-final last term to miss out on a first European trophy under Wenger.
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