Rescue efforts are under way, with officials saying that at least three people died in the multi-storey Melcom store.
Ten people have been pulled alive from the rubble so far.
Ghana’s President John Dramani Mahama said he had suspended his campaign for next month’s election.
The government has declared the area in the city’s Achimota neighborhood a disaster zone.
Vice-President Paa Kwesi Amissah-Arthur is at the site, co-ordinating rescue efforts at the building, which opened earlier this year.
Hundreds of rescuers are digging through the rubble, amid fears that dozens of people are trapped inside, our correspondent says.
Paa Kwesi Amissah-Arthur said the building collapsed shortly before the Melcom store was due to open for business.
Rescue efforts were being hampered by the large crowd that had gathered at the site, he added.
“There’s so much noise that it’s not possible to hear the trapped people asking for help,” Paa Kwesi Amissah-Arthur said.
“If we could move out of here, it could help rescue operations.”
In a statement on its Facebook page, Melcom said it had rented the building which housed its Achimota branch on a 10-year lease.
“This is indeed a very tragic incident,” it said.
“We are doing everything possible to see that help reaches those who need it. Our heartfelt condolences and deepest sympathies goes to the families of those who may have lost their lives.”
Customers often queue outside the store before it opens and officials say they believe about 50 people may been inside at the time of the collapse.
Eyewitness Ama Okyere told the AFP news agency she was very close to the shopping centre when the building came down.
“I had to run for my life. I was so terrified. I believe there are lots of people trapped under this because this is a heavily patronized shopping mall in the area.”
Family members have been trying to call relatives feared trapped beneath the rubble on their mobile phones, AFP reports.
Another witness, John Owusu, said he heard a bang before the building collapsed.
President John Dramani Mahama, in a tweet moments after the building collapsed, said: “My prayers are with the workers, shoppers and others who are trapped in the rubble of the Achimota Melcom building.”
He has cut short his campaign in the north of the country and is flying back to Accra.
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