Jenkinson, 22, joined Arsenal from Charlton in 2011 and earned his only England cap against Sweden 18 months later.
“I’m clearly seen as a big part of things here and hopefully I can prove my worth because it is an important year for me,” said Jenkinson, who joins Mauro Zarate, Enner Valencia, Cheikhou Kouyate, Aaron Cresswell and Diego Poyet in a new-look West Ham squad.
“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.”
Meanwhile, Southampton have completed the signing of left-back Ryan Bertrand on a season-long loan deal from Premier League rivals Chelsea.
The Saints had been looking for a full-back to replace Luke Shaw, who was sold to Manchester United for a fee of around £30 million earlier in the transfer window.
And it would seem Bertrand is just that, with the loan deal including an option to make the move permanent next season.
“I’m very happy to be able to bring Ryan in,” head coach Ronald Koeman told the club’s official website.
“I spoke to him on Monday afternoon and I was very impressed with his attitude and how keen he was to play games for Southampton.
“He is a top player who will bring a lot of quality to the team. He has experience at the highest level and is an exciting, attacking player, which is what I like to see in my full-backs.”
The 24-year-old added: “For me personally, this is another chance to play and a chance to join a fantastic club that’s moving in the right direction.
“With the set-up the club’s got, the manager it has coming in and the ethos it has, I thought this was a perfect match.
“The manager’s name speaks for itself, so to play under somebody like that will be good for me. His ethos and the way he plays matches my game.
“I had a lot of options and a lot of interest but, having met the people and spoken to the manager and seen what the club is about, in my mind this was the perfect decision.
“I’m looking to get games and to play regularly in the way and the formation that I like, and I’m sure that will benefit my game. I still think the best is yet to come from me.”