The man died at North Manchester General Hospital on Sunday after coming back from a trip to northern Italy at the end of February, his son told BBC Bangla.
He said his father, a British national originally from Bangladesh, had fallen ill "instantly" and was taken to hospital after going to a local health centre for a routine appointment.The rest of the family were told to self-isolate, he said.
The son, who was not named, said he rang the hospital regularly for updates on his father, who had arthritis and heart problems, and then received a phone call saying he had died.
He told BBC Bangla: "Obviously I could not believe it because two months ago this thing didn't even exist and today it took away my father.
"It took me quite a long time to process the whole thing that I'm not going to be able to see him any more."
The patient, from Greater Manchester, was the third person to die in the UK after testing positive for coronavirus.
A spokesman for the hospital said: "We can confirm that sadly a patient being treated for Covid-19 has died at our specialist regional infectious diseases unit at North Manchester General Hospital.
"The man in his 60s tested positive after travelling to Italy and had a number of underlying health problems.
"Our thoughts are with the patient's family who are being supported by our specialist bereavement staff."