However, the youngster is only on loan from Arsenal, and if the Daily Mail is to be believed, Arsene Wenger wants him back at the Emirates next season, leaving the Hammers short of a top level full-back.
There are many players who could do a job for the club, but few players possess the same skill set as Jenkinson. An attacking full-back, he covers a wealth of ground, is good with the ball at his feet, and is not scared to take a man on to cross the ball.
This kind of attacking player is a scarce commodity on the kind of budget that West Ham can splash out on a player, but there is one available star that the Hammers should go all out to sign this summer.
Kieran Trippier has excelled this term for Burnley, but the club saw themselves relegated back to the Championship after just one season in the top flight. However, the likes of Danny Ings, George Boyd and Trippier himself have done enough to secure moves to Premier League clubs, and it is the former Manchester City right-back that the Hammers should target.
He is young - at 24 he is only one year older than Jenkinson - and possesses all the same abilities as the Arsenal man.
Highly energetic, and able to cover a lot of ground, Trippier has played in all 26 games for Burnley this term, and has registered four assists. He plays 1.8 key passes per game, 2.4 crosses per game - the highest in the entire division - but is still able to impress with his defensive duties.
Trippier fulfils 1.9 tackles, 2.1 interceptions and 4.2 clearances per game, whilst also only giving away 0.4 fouls per game. He outperforms all the West Ham full-backs in both attacking and defensive work, and would be available for cheap following the Clarets relegation.
Trippier is sure to have many clubs interested in his services this summer, but with a deal for Carl Jenkinson looking unlikely, the Hammers should do all they can to secure the 24-year-old full-back - as he is all the club need to fill the void left by the departing Arsenal loanee.