Downing enjoyed an excellent campaign last term and enhanced his claims for a more regular spot in the England side.
His performances did not go unnoticed and speculation has been mounting that he could be set to join one of Boro's Premier League rivals.
Liverpool are reportedly leading the chase and Riggott admits he can understand why Downing might be tempted.
"He's still a Middlesbrough player and we want him to stay, obviously all the fans want him to stay," Riggott told Sky Sports News.
"He's a top class player and blaming him or not blaming him, it's not really for me to say.
"But it's obviously a pretty big draw, a club like Liverpool or somewhere like that."