As a result, a transfer in the summer looks far more likely, and West Ham are hoping that Jenkinson will tell the Gunners of his desire to leave.
Doing so may be a wrench for the 22-year-old, who was a boyhood fan of the side, but his development would arguably be better served with Sam Allardyce's side.
Jenkinson looks unlikely to break into Arsenal's first-team - with Mathieu Debuchy, Calum Chambers and Hector Bellerin all ahead of him in the pecking order for the right-back berth - but he is a key player for West Ham.
The England international has already made 22 appearances for the Hammers, where he has established his place in the starting line-up, and this total is only set to increase by the end of the season.
Allardyce has identified Jenkinson as a player who he can plan his team around for the years to come, and he is probably hoping that he leaves Arsenal with no choice but to agree to compromise and sell him.