The 29-year-old winger failed to make an impact during his two seasons at Anfield after joining the Reds from Aston Villa in 2011.
With his chances limited under manager Brendan Rodgers, Downing admits that he had little choice but to leave Merseyside.
"I didn't want to sit around because Brendan will want to keep on spending and reinforcing and wanting his own team," he told The Telegraph. "So I had to make a decision and Sam Allardyce gave me the opportunity.
"I was happy to make that move. It's a great club, a club on the up and finished 10th last year and we want to finish as high as we can."
Downing has made five appearances in all competitions for West Ham this season.