The 22-year-old joined Arsenal from Charlton Athletic in 2011, making 57 appearances as back-up to first-choice right back Bacary Sagna.
After Sagna joined Manchester City in June, Arsenal replaced him with Newcastle United's Mathieu Debuchy and Southampton's Calum Chambers has also joined the club, so Jenkinson was allowed to seek a loan move by manager Arsene Wenger.
"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," Jenkinson told the east London club's official website.
He becomes West Ham's sixth signing of the close season following the arrival of forwards Mauro Zarate and Enner Valencia, Senegal midfielder Cheikhou Kouyate, defender Aaron Cresswell and midfielder Diego Poyet.
West Ham begin their Premier League campaign at home to Tottenham Hotspur on August 16.