The Hammers have suffered with injuries this season and Jenkinson was the latest to be ruled out long-term after suffering what West Ham described as a 'significant' knee injury in the 2-2 draw with Manchester City last time out.
Slaven Bilic has only just signed exciting young full-back Sam Byram from Leeds United who replaced Jenkinson with aplomb after he limped off 10 minutes into the City clash.
But with Jenkinson likely to be out for much of the remainder of the season it leaves Bilic short of options given James Tomkins has been playing out of position to fill in there for much of the season already.
Arsenal right back merry-go-round
However, the report also says Arsene Wenger may pull the plug on Debuchy's exit as it would leave him with only Hector Bellerin as they aim for a first title in 12 years.
Debuchy would undoubtedly prefer a move to West Ham, though, with Bilic's side flying high in the Premier League and aiming for a European place.
The Hammers are also on the brink of their big move to the 54,000 seater Olympic Stadium in the summer and the Frenchman would not have to relocate either if he moved across from North to East London.
Jenkinson has been out of sorts this season and was never a realistic summer target for West Ham given he was recently handed a new four-year contract by Arsenal.
So West Ham would most likely look to loan Debuchy for the rest of the season with an option to buy in the summer, a system favoured by the club's co-owner David Sullivan.