It has been reported by the British newspaper that Arsenal want £2m as loan fee to send the England international defender back to their London rivals on a temporary basis.
The right-back played on loan for the Hammers during the recently concluded 2014-15 campaign and gave a good account of himself.
Jenkinson made 32 appearances in the Premier League for West Ham, creating 22 chances in the process.
The 23-year-old right-back had a pass accuracy of 78% and a shot accuracy of 60%, and he won 22 headers, 11 dribbles and 26 tackles.
Former Charlton Athletic defender Jenkinson progressed well at the Boleyn Ground, but he will find it tough to break into the Arsenal first team next season.
Mathieu Debuchy, formerly of Newcastle United, is the Gunners’ undisputed number one, while youngster Hector Bellerin is his deputy; Calum Chambers can also play at right-back.
It would be best for the development of Jenkinson that he stays at West Ham next season, but it remains to be seen if the Hammers are willing to pay the £2m loan fee that Arsenal reportedly want.