Arsenal are willing to loan out Jenkinson following the recent arrivals of Mathieu Debuchy and Chambers meant the England international's first-team chances limited this season.
Gunners boss Arsene Wenger sees Jenkinson as part of Arsenal's long-term future, but is happy for him to go out on loan to get regular first-team football under his belt.
There has been no shortage of takers for Jenkinson with West Ham, Hull City, Swansea City and Sunderland all thought to have enquired about the 22-year-old.
However, West Ham are in pole position to land the former Charlton man after holding talks with the player earlier on Monday.
West Ham boss Sam Allardyce is thought to be an admirer of Jenkinson and sees him as a valuable addition to his defensive options next season.
Jenkinson is happy to go out on loan to play regularly next season as he looks to win his place back in the England squad and the move to Upton Park is expected to be sealed within the next 24 hours.