The Irons had hoped to sign the 23-year-old permanently from their London rivals but Gunners boss Arsene Wenger is reportedly keen to keep Jenkinson in the long-term.
Flying full-back was a revelation
The flying full-back was a revelation for the East Londoners last season and many thought he could force his way back into Wenger's plans as a result.
But with Arsenal well stocked in the right-back berth with the likes of Hector Bellerin bursting onto the scene, they can afford to let Jenkinson leave on loan.
In fact, Jenkinson's form for the Hammers has actually made life even more difficult for the club to get him signed up as it has alerted a number of top clubs at home and abroad to his quality and availability.
Hammers are going for a Song
Slaven Bilic's side are expected to complete an Arsenal reunion with the Daily Mail also reporting that another of West Ham's loan stars from last season, Alex Song, is also set to sign - this time permanently.
Ex-Gunner Song, who has been linked with a move to the likes of Manchester City, Liverpool and Chelsea, started last season like a house on fire following his shock loan move from Barcelona.
But like the team, his form faded after Christmas leaving question marks over his future.
The Mail's report says personal terms have been agreed, though, and shows video footage of Song in pre-season training with the Hammers.
We're Gunner need a bigger... squad
Should West Ham announce the signings in the coming days as expected, it would be a major boost as they start competitive football before any other Premier League side thanks to their involvement in the first round of qualifying for the Europa League.
With the potential to play as many as 23 extra games in the competition, the Hammers will need a big squad.
They have already added goalkeeper Darren Randolph and midfield stars Pedro Obiang and Dimitri Payet to their ranks with the promise of more to come.