Jenkinson's first-team chances at Arsenal were limited following the recent arrivals of Mathieu Debuchy and Calum Chambers and boss Arsene Wenger is happy for him to go out on loan to get regular first-team football.
There have been a number of clubs interested in the England international with Hull City, Swansea City and Sunderland all thought to have enquired about the 22-year-old, but Hammers boss Sam Allardyce has won the race to sign him.
Jenkinson told West Ham's official website: "I'm clearly seen as a big part of things here and hopefully I can prove my worth here because it is an important year for me.
"I was flattered to have a lot of options but, deep down, if I was being honest with myself West Ham was always the place I wanted to go to.
"It's one of them where I wanted to do the right professional thing and consider all the options, but in the back of my mind it was always West Ham and I'm just so glad it's all worked out how I wanted it to in the end.
"It will be brilliant to play here. I've got a lot of friends who are West Ham fans and I know how good the supporters are here.
"I've played here a couple of times so I know exactly what it is all about, so it's going to be fantastic to be a part of that,
to be honest. You get a great atmosphere here and the fans are great, so I just can't wait to get a West Ham shirt on and get on the pitch and play. This is an exciting time for me, it really is.
"There is an English group of lads here, which is always great for an English lad, and the players who have been brought in are top-drawer, so there is every reason to be optimistic about this season. That was a big reason I signed here."