The Teessiders face losing current No 1 Mark Schwarzer when his contract expires in the summer.
And though Coupet signed a new deal with French champions Lyon last season, the agreement contains a £3 million release clause with the 35-year-old - first choice for Les Blues - now keen to leave his homeland.
"He is seriously considering his options," his agent Frederic Dobraje said.
"His desire to move on, which became apparent last season, has become even greater this season. It is a desire which Greg has held for some time."