Downing, who has 34 caps for England, has signed a four-year contract with the London club and is Sam Allardyce's fifth summer signing.
"I'm very excited," he said. "It's a great opportunity for me being here, it's a great club and I'm really excited to get going.
"I'm ready and I'm fit for the first game against Cardiff City. I've played a decent few games for Liverpool in pre-season, so there will be no problems on that front.
"It's all happened quickly, but I'm really excited and hoping to play a part on Saturday."
The 29-year-old moved to Anfield from Aston Villa for £20 million in July 2011 and made 91 appearances, scoring seven goals, for the Reds.
The England international struggled to cement a place in the first team, particularly in his favoured forward position, and Liverpool boss Brendan Rodgers informed the former Middlesbrough man he was free to leave last season.
West Ham boss Allardyce will be hoping Downing can provide the crosses to help former Liverpool striker Andy Carroll thrive at Upton Park.