As reported by Sky Sports, the deal for Ramsdale could potentially rise to £25million if performance-related add-ons are met.
The report states that negotiations are continuing to take place regarding other aspects of the deal but there is hope that the move can be finalised.
Arsenal have informed Ramsdale that he can leave if the Gunners receive a suitable offer for him.
Southampton have agreed a fee of £18million with Arsenal for goalkeeper Aaron Ramsdale
Bijlow had been poised to sign for Southampton but the deal didn't take place after a problem with his medical.
Ramsdale has fallen behind David Raya in the pecking order at Arsenal, with his last game for the club coming against Brentford in March.
England international Ramsdale joined Arsenal in 2021 and has played 89 times for the club.
He previously played for Sheffield United and Bournemouth.