The full-back has impressed with his displays at the Boleyn Ground since joining and has already become a key part of the Hammers' first-team squad.
However, Allardyce is reportedly unwilling to match the reported asking-price, due to its ecessive nature.
West Ham had been hopeful of concluding a deal in January, with Jenkinson's chances of breaking through in North London looking limited, but they will have to negotiate with the Gunners if there's to be any type of agreement.
Jenkinson appears to be the third-choice for the right-back berth at the Emirates with Mathieu Debuchy and Calum Chambers both ahead of him in the pecking order.
With Hector Bellerin also coming through, there are serious question marks over whether the 22-year-old will be able to make the grade at the club he supports.
At West Ham, however, he looks a player who Allardyce can trust to be a member of his starting line-up for years to come.
Jenkinson still needs to develop, but he has played 13 times already this campaign and his energetic displays have been a key part of West Ham's success.
If the Hammers are able to sustain this form and claim a European spot then the increase in finances as a result, may mean that a deal for Jenkinson becomes more likely in the summer, but any move before that appears to be a long-way off.