Downing penned a new five-year contract in February in a bid to end speculation linking him with a transfer away from the Riverside.
However Downing has admitted he may look elsewhere if his current club, who finished 13th in the Premier League last season, fail to eventually qualify for Europe's elite competition.
Downing: "It is unsettling when you are younger, definitely. Obviously if there are big clubs interested in you it is flattering and sometimes you are tempted.
"But at the moment I'm focused on doing my best for Middlesbrough.
"My mates are big Boro fans and they obviously want me to be at Middlesbrough for all my career, and if we become a top team, a Champions League team, then there would be no question about going.
"But if we don't eventually make that step, then one day I might have to make that decision. It's an important couple of years for me.
"It might be good one day to do that, maybe move on and try something different, I think every player likes to try something new in their career and I'm no different."