Clattenburg has been accused of making racist remarks towards some Chelsea players during the controversial match, which ended 3-2 to United, after two Blues players were sent off.
A formal complaint has been lodged with the Football Association and now the Society of Black Lawyers wants the police to investigate further.
"The Society of Black Lawyers wishes to record the incident that took place at Chelsea FC,” read a letter from the organisation's chairman Peter Herbert to the Metropolitan Police.
"Our information is that racist remarks were directed at John Mikel Obi and at Juan Mata. If so, that is wholly unacceptable in any circumstances.
"Although this matter may be investigated by the FA, it is appropriate that the MPS independently see if a racially-aggravated offence has occurred."