Mito Kaur, 63, picked up the mucor fungus at the Queen Elizabeth University Hospital, where two patients recently died after contracting an infection found in pigeon droppings.
She passed away at 2 am this morning after her life support machine was switched off.
Since December, five people have died after contracting a hospital infection in the Glasgow area - the death is the latest in a series of scandals to hit the Queen Elizabeth University Hospital.
Last week a damning inspection report pinpointed poor standards of cleanliness at the £842m hospital which opened in 2015.
READ MORE: Scotland's biggest hospital needs more staff to meet hygiene standards The inspection was ordered after patients became infected with a fungus linked to pigeon faeces.
One of those was a 10-year-old boy who died in December after contracting cryptococcus.
An adult who had the same infection died the following month but the hospital said cryptococcus was not a factor in her death.
Mrs Kaur's lawyer Aamer Anwar issued a statement saying he has contacted the Procurator Fiscal's Office advising of the family's concerns and their desire for a fully independent investigation to take place.
He added: " I understand that the Crown Office and Procurator Fiscal have instructed the police to obtain a full report, following which an independent post-mortem will be carried out.
"In recent days an unannounced inspectorate report into the management of infection and control following two previous deaths was published, this was a devastating indictment of the culture that existed at the QEUH.
"The family are deeply grateful to the many staff who tried so hard to save their mother’s life, but they remain highly concerned at the role of senior management and demand a full robust and independent inquiry take place.
"At this stage it would be premature to say anything further about the cause of death, but I would simply ask that the privacy of a grieving family is fully respected"
The recent inspection report found 300 repair jobs were waiting to be done at the hospital, but there was no evidence of a plan to complete them.
It also said the emergency department had not been properly cleaned.
This included body fluid and grime on toilet seat hinges in the reception and patient areas of the department, as well as dusty and gritty floors and what appeared to be blood on two patient trolleys that were ready for use.
Mrs Kaur was taken to hospital after contracting flu which then developed into pneumonia.
She then picked up a mucor fungus and the family was told it was potentially fatal.
The infection caused brain to swell and, after rallying briefly, her condition deteriorated again and her heart became inflamed.
By February 6, Mrs Kaur had deteriorated and two days later the family were told she would certainly die and should consider when they wanted her to have life support switched off.
Before she had contracted flu, Mrs Kaur had been a happy healthy mother of four and grandmother of nine.
Her son Bobby said she was the “heart” of the family and her children were so devoted to her they have been taking shifts to stay at the hospital.
Mrs Kaur, a shop worker, was planning to retire in March to spend more time with her family and she considering a trip to India.