Tottenham and Liverpool are chasing the England winger ahead of the January market.
Asked if he could still achieve his ambitions with his hometown club, Downing answered: "I hope so, I signed a long term deal earlier this year and my only aim at the minute is to try and get into Europe with Middlesbrough.
"This is my hometown club, I started here and it's the only club I've played for apart from a spell on loan at Sunderland."
"There is always going to be speculation with players and you've just got to blank it out because if you concentrate on what the press is saying your form will dip and that doesn't help you or the team. So I just want to concentrate on playing well for Boro."