Jenkinson, 23, is currently on a season-long loan deal at West Ham having joined from Arsenal last summer, with the Gunners keen to give him experience.
Yet with Hector Bellerin emerging at Arsenal, it seems that the Gunners are happy to let Jenkinson leave permanently this summer, and after a superb season at Upton Park, the Daily Mail believe that West Ham are willing to pay around £10m to land him on a full-time basis.
However, Jenkinson's impressive performances may attract other clubs, with The Sun reporting last week that Liverpool now had their eyes on Jenkinson as a replacement for out-of-contract Glen Johnson.
Johnson looks set to leave Anfield on a free transfer this summer, but in a surprise merry-go-round, he may end up at West Ham as a replacement for Jenkinson.
Johnson, 30, came through the famed West Ham academy having joined at the age of 15, but left just three years later to join Chelsea.
Spells with Portsmouth and now Liverpool have followed, but with his contract at Anfield set to expire this summer, he would be able to return to Upton Park for free.
The Hammers' priority will of course be keeping Jenkinson, but if the Daily Mail are to be believed, Johnson is their contingency plan ahead of the summer transfer window.