House of Fraser said the planned closures come as part of a company voluntary arrangement (CVA) which would affect 31 stores across the country.
More than 160 jobs could be lost at Chichester’s House of Fraser.
The company said shops earmarked for closure would remain open until early 2019.Teresa Cash, who lives in Littlehampton and was visiting Chichester with her son for a day of shopping, said: "I think it's very sad.
"It's sad for the city of Chichester to lose a store that's been there so long in one form or another. What they will do with it, I don't know.
"It's a sad day for the people who work for House of Fraser.
"I hope they can put something in there which will be good for the town. I suppose it reflects the changing face of the high street."
Pete Newman, 75, and his wife Anne Newman, 73, from Hayling Island, who had just visited the store, said they were disappointed by the news.
They believed the closure would have a big impact on Chichester as a shopping destination.
"It's the only really big department store, they've got all the different franchises," Mr Newman said.
"I think it probably will make a difference. Trouble is, these kind of stores bring in people and smaller stores get the spin off."
But Brian Oertel, who was visiting from Bedford with his wife Valerie, said the closure would not deter the couple from coming to Chichester, which they agreed was a 'lovely' place to shop.
However, he said: "It's opposite the cathedral. The effect will be, you get off the bus and you're going to see this huge empty building. It's going to immediately demoralise people."
He said the store was 'a British institution' but said: "It's great for Christmas, nice for gifts and things, but it's not where the everyday shopper goes. They've not kept up with the trends."
Lena Sutton who works in sales at the Sofa Workshop a few shops down said: "I think it's a shame.
"Some people want to shop in House in Fraser and come to Chichester particularly for that. They tend to get an older customer."
On the effect it would have on the furniture store, she said: "I think another place closing down is never good for business. It always helps if you have competition."
Since the closure was announced, rumours have been circulating on social media about a Primark moving into the building.
But a Primark spokesperson told the Observer: “Primark does not comment on speculation or plans regarding new store openings.”
Chichester resident Lyndsay Vass said a Primark would 'be a lot cheaper and probably busier'.
He said it was inevitable that House of Fraser would close, adding: "It's too expensive. They haven't moved with the times.
"People don't want to pay the prices anymore."
Tim Matthews, who sells homemade hot dogs and burgers from a stall in West Street, said: "I'm shocked at another shop closing.
"The city's footfall has dropped rapidly. I've seen it drop over the last two years at quite an astonishing rate really.
"It's sad, it feels like it won't be long before all the major shops go."
Geoff Holland, who visits Chichester every Thursday from Fishbourne and often goes to House of Fraser, said: "I think it's a great shame we're losing it. I'm very surprised.
"It's bound to have some impact, I can't help feeling.
"If anyone wants to go to a shop like that they're going to have to go to Portsmouth or Southampton or Worthing."
Ellen Altria, who has lived in Chichester for 25 years, had just visited the store and said she 'really wanted' it to stay.
"People say it's expensive but I can always find something," she said. "It will be a sad day when it goes."
Donna Treagust, a sales associate at Orvis in South Street said: ""I was surprised but in a way not surprised.
"It's a shame it's going, it's part of Chichester. Customers have been in here saying the same thing."
She said it might have an impact on Orvis, as the store would often send shoppers over for Barbour items if they did not have them in stock.
"I think it's quite sad for the staff that work there," she added.
A spokesperson for House of Fraser said: “House of Fraser’s store in Chichester is among the 31 sites that has been identified for closure.
“All 166 staff members employed directly by the company or its concessions have been notified and will be supported throughout this process.”
Frank Slevin, chairman of House of Fraser, said: “The retail industry is undergoing fundamental change and House of Fraser urgently needs to adapt to this fast-changing landscape in order to give it a future and allow it to thrive.
"Our legacy store estate has created an unsustainable cost base, which without restructuring, presents an existential threat to the business.
"So whilst closing stores is a very difficult decision, especially given the length of relationship House of Fraser has with all its locations, there should be no doubt that it is absolutely necessary if we are to continue to trade and be competitive."