The e-petition calls for the Brighton and Hove local authority to take a stand and not offer "hospitality" or use royal or aristocratic titles which are "are arbitrarily and unfairly acquired".
Campaigner Charles Ross has accumulated 3,881 signatures meaning the motion will have to be discussed at a council meeting today.
As the council does not have the authority to remove titles given by the Queen, Mr Ross calls on officials to stop calling them the Sussexes in council documents.
"We the undersigned petition Brighton & Hove Council to reject the usage of the titles "Duke of Sussex" and "Duchess of Sussex" by the individuals Henry ("Harry") Windsor and Rachel Meghan Markle as morally wrong and disrespectful to the county of East Sussex," Mr Ross wrote.
"As residents of Brighton and Hove we call on Brighton and Hove Council to not refer to these individuals by such titles which we believe to be entirely non-democratic and symbolic of the oppression of the general public by the wealthy elite.
"Neither will Brighton Council invite or entertain these individuals nor afford them any hospitality or courtesies above and beyond that of an ordinary member of the public."
Mr Ross also said the council needs to take a stand and added: "The petition aims to establish a precedent that Brighton and Hove Council will no longer afford official hospitality to those with Royal or aristocratic titles nor make usage of those titles in official documents as these titles are arbitrarily and unfairly acquired."
The Duke and Duchess of Sussex were well received on a trip to the county last October.
Hove MP Peter Kyle praised them at the time and said the couple reflected the diversity of Brighton.
Brighton and Hove Council will comment after the petition had been debated at tonight's meeting.
