The woman said that she and others had contacted Barlinnie to raise concerns.
The SPS could not say how many prisoners had been affected but it is understood those in A and E halls had problems with their supplies.
A Scottish Water spokesperson said: “We can confirm there are no issues affecting the Scottish Water network in the area.
“The matter relates to an internal issue within HMP Barlinnie and we are supporting the Scottish Prison Service as it works to resolve it.”
Last year the late 19th Century and early 20th Century sections of the Glasgow prison were recognised by Historic Environment Scotland (HES) as a Category A-listed building.
Construction of Barlinnie started in 1880 and the site opened in 1882.
It was the first new prison built in Scotland following the nationalisation of the prison system, with the Prison Act of 1877.
Barlinnie was known for being a notoriously tough prison, where prisoners carried out hard labour, breaking rocks from a local quarry.
It also served as a home for capital punishment, with 10 men executed by hanging at the site between 1946 and 1960.
In recent years the prison has been plagued by issues of overcrowding.
The Scottish government has committed to building a new prison to replace Barlinnie and construction is under way.
It is due to be completed in 2028 at a cost of nearly £1bn.