Arsenal boss Arsene Wenger is thought to be open to the idea of loaning out Jenkinson next season so that he can play regular first-team football.
There is no shortage of suitors for Jenkinson, with West Ham, Swansea, Sunderland and Hull all interested in taking the former Charlton man on loan.
The quartet are ready to offer Jenkinson, who made 22 appearances last season, regular first-team football next term as they look to bolster their defensive options.
Hull boss Steve Bruce confirmed to Sky Sports News that he has enquired about Jenkinson's availability.
"I had a conversation with Arsenal about four or five weeks ago to see if it was possible. And apparantly so, it might be," he said.
"We'll look at that situation but we haven't had confirmation from Arsenal. I would hate to speak out of turn but we are led to believe the kid might be available - and if he is, then of course we would be interested."
Jenkinson is expected to find his first-team opportunities limited at Arsenal following the signing of Mathieu Debuchy and the imminent arrival of Calum Chambers at Arsenal.
Chambers is set to arrive from Southampton with the 19-year-old right-back viewed as a possible central defender in the future by Arsenal.
England international Jenkinson remains highly regarded at the Emirates, but Wenger feels he would be best served by going out on loan to another Premier League club next season to continue his development so that he is ready to stake his claim for a regular spot in the Arsenal side upon his return.
Jenkinson sees his long-term future at Arsenal, but knows the opportunity of first-team football in the Premier League next season will boost his chances of reclaiming his place back in the England squad.