The piece features a stencil of a girl firing a slingshot of red flowers made with spray-painted ivy.
Banksy published a picture of the work on his Instagram page at midnight on Valentine’s Day.
Kelly Woodruff, whose father owns the property in Barton Hill, said they now wanted to protect the “special” work from the approaching Storm Dennis.
“As it’s Valentine’s Day, it’s really special and to have it in 3D with the flowers is incredible,” said 37-year-old Ms Woodruff.
“There’s so many people coming and enjoying it, taking pictures, it’s fantastic.
“There’s been a lot of debate if it is a Banksy or not. Most people I’ve spoken to think it 100% is, and they’re naming it the Valentine’s Banksy.
“It’s incredible and beautiful.
Flowers placed on the road sign as part of the artwork had already been stolen, said Ms Woodruff, along with a protective plastic screen they placed over it on Thursday evening.
Ms Woodruff said her family had contacted Bristol City Council’s street maintenance department but it was unable to offer any advice or help.
“We’re not sure where to go from here. We’re going to get some advice [about protecting the work],” she added.
The council has been approached for comment.