Bolasie, on loan at the Sky Bet Championship club from Premier League Everton, revealed the abuse he had received in a Twitter post on Wednesday.
In a statement, Boro said the abuse “demonstrates exactly why more has to be done to identify the perpetrators of such comments and bring them to justice.”
The club confirmed the matter has been reported to the relevant authorities, by both Bolasie and the club, with the 31-year-old receiving Middlesbrough’s full support.
The statement continued: “It is of great concern to the club that we are witnessing behaviour of this type and it cannot be allowed to continue.”
“As our players said when responding to the debate on taking the knee we now need to see demonstrable actions from those who have the power to implement them rather than further gestures. Actions must speak louder than words.
“We repeat that we will not tolerate abuse of any form and any individuals who are identified to us will not only face the prospect of criminal charges but will also be banned from the Riverside Stadium.”
Instagram is investigating the incident.
Bolasie tweeted on Wednesday: “Something seriously wrong with people…Keyboard warriors. Still yet to meet a person who had this energy when they saw me.”
Bolasie’s parent club also condemned the attack on their player, with a statement from Everton reading: “Everton has a zero-tolerance approach to any kind of racism or discrimination. There is no room for racism in football or society.
“By working together, we can fight discrimination wherever it exists. Everybody has a part to play and social media abuse should never go unreported.
“We also urge social media platforms to strengthen measures to prevent this kind of behaviour. Through challenging it and reporting it, we can all play a part in changing it.”
Democratic Republic of Congo international Bolasie joined Boro in January for the remainder of the season and has made four appearances for the club to date, the most recent of which was Saturday’s 2-1 defeat at Swansea.