Home fans allegedly chanted racist abuse towards Arouna Kone after he celebrated the opening goal with Aaron Lennon in Everton's 3-0 victory at Brunton Park.
"We are investigating a report and will be working with the club to ascertain what happened," a police statement said.
The official Twitter account of Cumbria Police said: "Racism will not be tolerated in Cumbria. Anyone with information on any racist chanting at the @officialcufc game please report it."
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Speaking after the match, Carlisle manager Keith Curle confirmed that the incident was reported to referee Lee Mason immediately after the opening goal.
Curle ordered an announcement to be made over the public address system to try and prevent any further incidents.
"It was Kone and Lennon -- the referee came over after the first goal and said some comments were made and some of the players heard some comments," he said.
"I spoke to the fourth official. An announcement was made to say it's not tolerated within the boundaries of any football club and not accepted within the stadium."
Everton manager Roberto Martinez added: "I thought it was dealt with in a good manner.
"Straight away the Carlisle manager Keith Curle and myself, we agreed to make an announcement. I thought the referee dealt with that well.
"The worry was it would be a 90 minutes of abuse but it was just a reaction to a goal being celebrated."
(Reporting by Mike Collett; editing by Martyn Herman)
