The Champions League winning club says it wants to protect both its fans and itself from fake Liverpool goods being created by taking ownership of the name.
It says the rules would only be enforced in a football context, so road signs, businesses and tourist attractions wouldn't be affected.
"At the end of the day, it's only a football club. The city is bigger than a football club," one fan told ITV News.
"I don't think it's fair" and "Why should it just belong to a football club when it's about the people, the culture, the landmarks?" responded two others.
One fan, not opposed to the idea, stated: "Chelsea have done it. Southampton have done it. Why not?"
Where does that leave other football clubs bearing the word Liverpool in their name?
"We took the name City of Liverpool from the community as a community-owned club," Peter Furmedge told ITV News.
"Suddenly that would belong to Liverpool football club, not the city itself."