The 22-year-old has been informed that he is surplus to requirements at Emirates Stadium, with Mathieu Debuchy having arrived from Newcastle and Calum Chambers set to sign.
The 22-year-old has been told he has no future at the Emirates Stadium, with Mathieu Debuchy set to be the first-choice right back following his arrival from Newcastle, while Southampton defender Calum Chambers is on the verge of completing a switch to North London.
West Ham is Jenkinson's preferred destination and Sam Allardyce's side could yet hijack the deal if it comes up with the cash in the next couple of days.
MORE: Beautiful fans in Brazil | Barcelona's new bright kits | Juve's jerseys
Sunderland and Swansea are also interested but it is Hull that is the front-runner for the England Under-21 defender ahead of the start of its Europa League campaign, which kicks off Thursday against Slovakian minnow FK Dukla Trencin.
Jenkinson, who made his England debut in 2012 but has struggled to become a regular in Arsene Wenger's first-team setup at Arsenal, is now set to discuss personal terms with Hull, with Steve Bruce confident of tying up a deal within the next 48 hours.
The former Charlton Athletic defender will be the Tigers' fourth signing of the summer window, with Jake Livermore, Robert Snodgrass and Tom Ince having all arrived ahead of the 2014-15 campaign.
Sheffield United defender Harry Maguire could also join in the coming days, with the League One side confiming Monday that it has accepted a bid from Bruce's side for the 21-year-old.