Official documents show there were two cases of the MDRO (multi-drug resistant) infection pseudomonas aeruginosa at Glasgow Royal Infirmary.
The cases were 'red flagged' by health bosses last month, the most serious outbreak category.
NHS Greater Glasgow and Clyde confirmed one of the patients had contracted the bug due to 'cross contamination' in the intensive care unit.
The health board said one patient, who had serious underlying health conditions, had died during the outbreak, but did not issue further details.
Pseudomonas aeruginosa rarely affects healthy patients and can cause a wide range of infections, particularly in people with a weakened immune system.
According to UK Government guidelines, infections are sometimes associated with contact with contaminated water.
Water sampling tests are said to have ruled out any trace of the bacteria.
In hospitals, the organism can contaminate devices that are left inside the body, such as respiratory equipment and catheters.
Some varieties of the bug are said to be resistant to nearly all antibiotics.
Professor Hugh Pennington, Emeritus Professor of Bacteriology at Aberdeen University, added: "It is an ICU pest when it contaminates equipment and has been a significant problem (pre-Covid) for patients being ventilated."
NHSGGC said blood cultures had detected a typing match between the affected patients.
Tests of water, gloves and all shared equipment were negative but the board said investigations were ongoing.
A spokeswoman for NHSGGC said: "Our condolences go out to families and friends who have lost a loved one.
"While we cannot comment on individual cases, we can confirm incident management teams were established in relation to these incidents and swift action was taken as a result.
"We have strict infection prevention and control procedures in place and all staff follow national guidance.
"These incidents have been reported to our board as part of our robust management and governance of infection prevention and control processes in our hospitals."
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