The scene was disturbing ... just minutes after players took a knee to show support of the Black Lives Matter movement -- the plane circled around Etihad Stadium in Manchester, England.

Reports say the banner was visible for several minutes ... and players -- who have all decided to replace their last names on their jerseys with the words "Black Lives Matter" -- said afterward they were sickened by the message.

Now, cops have vowed to launch a probe into the matter ... writing in a statement Tuesday morning, "We are making enquiries to establish the full circumstances surrounding this incident."

"We will then be in a position to make an assessment as to whether any criminal offences have taken place."

Burnley's officials also condemned the banner -- which was allegedly paid for by a Burnley fan -- writing in a statement of their own, "This, in no way, represents what Burnley Football Club stands for."

"We will work fully with the authorities to identify those responsible and issue lifetime bans."