Downing recently dropped down from the Premier League to the Championship, completing a return to home-town club Middlesbrough.
Following the Englishman's departure, though Bilic has expressed regret over the loss, as per BBC Sport. He said: "I didn't want him to go, he's a quality English player and I know how hard it is to get the quality ones. Of course I rate him, I tried to persuade him to stay with us.
"I know him from the national team when we played England and he was brilliant in the first half of last season. Maybe I could have stopped him, and all that, however it is not what we wanted to do. He wanted to leave."
The 30-year-old, who played almost 250 games for Boro, left the Riverside for Aston Villa in 2009 before moving to Liverpool. After a deeply unsuccessful spell on Merseyside, he made the move to Upton Park, where he made 37 league appearances last season.
He emerged as the Hammers' key player as the club finished in 12th place, scoring six goals and fashioning 85 key passes - more than any other player at the club and the fourth-highest total in the entire Premier League, behind only Eden Hazard, David Silva and Cesc Fabregas.
This summer's transfer activity, though, may have spelled the end for Downing, with Dimitri Payet joining West Ham, while his move to Middlesbrough may potentially free up space for even more additions at Upton Park.