The Englishman was also the subject of a bid from Hull City but opted to remain in London - as Goal revealed on Tuesday - having fallen down the pecking order at the Emirates following the signings of Mathieu Debuchy and Calum Chambers.
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He added: "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.
"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."
Jenkinson made 14 Premier League appearances last season, scoring once and providing two assists.