With fellow relegation strugglers Sunderland due at the Riverside on Saturday, Pogatetz said: "I wouldn't expect anybody to boo him because he is a local lad who always performs.
"All that matters to the club is that we get the three points. It shouldn't be about Stewart Downing and his transfer request.
"Stewart has been a good servant for many years. He's never let anybody down. We need the fans to get behind us."
Pogatetz spoke out after Boro rejected a £14.5 million offer from Spurs for Downing as the White Hart Lane club's efforts to sign the England winger intensified even though his transfer request was rejected.
"I was surprised to hear that Stewart had put in a transfer request," Pogatetz said. "It's always disappointing when someone wants to leave the club. I hope he stays because he is one of our best players."