Chairman Sharon Brittan said she was “sickened to the core” by the abuse, and called on social media platforms to make users accountable for their actions.
Bolton confirmed they are “providing support” to Kachunga, who joined on a two-year deal last month after a season at Wednesday, and have reported the matter to Greater Manchester Police.
Brittan said: “Racism and hatred have no place at all in society and I was sickened to the core on hearing this disgusting abuse directed at one of our players.
“I have spoken with Elias and his partner and they are understandably upset by these messages.
“My immediate thoughts are with them, as well as his team-mates who are repulsed by the language that was used.
“We as a club stand firm in saying that racism and discrimination will not be tolerated and we have reported this to the police and relevant authorities.
“I have also spoken with the EFL today regarding future steps we can take together to eradicate this hatred from our game.
“I will be relentless in making sure that everything is done to help identify and prosecute the perpetrator.
“We will not rest until positive action is taken and social media companies take the necessary steps to eradicate unsolicited hatred from their platforms.”