According to reports in France, the north east club believe the Danish forward could be the man to help fire them back into the Premier League.
Braithwaite scored 11 goals in 34 Ligue 1 games for Toulouse last term and will be allowed to leave if an appropriate offer comes in for him this summer.
It was expected to come from either West Ham or Southampton, who have both been tracking the 26-year-old.
However, he may be headed for England's second tier should Middlesbrough firm up their interest with an official bid.