William Chapman, 24, was mauled to death at the park near Doncaster on Saturday.
In a statement, the trust said following Chapman's death it had identified two possible incidents of staff "accessing or attempting to access incident/medical information" about the case.
It said an investigation had been launched "to establish the full circumstances surrounding these potential breaches".
Chapman died as a result of "multiple traumatic injuries and animal interaction" after entering the tiger's enclosure at the park, an inquest into his death heard on Thursday.
In a statement, a spokesperson for Yorkshire Ambulance Service said Chapman's family was aware of the potential breach and offered their condolences to his relatives.
They said: "We believe there may have been two instances involving two members of staff accessing or attempting to access incident/medical information following this incident without a legitimate reason to do so.
"We take the confidentiality and security of patient information extremely seriously and have robust systems in place to protect information, and this concern has been identified as a result of our proactive audits.
"The trust has commenced an investigation to establish the full circumstances surrounding these potential breaches and the actions of these individuals.
"We have also reported this matter to the Information Commissioner's Office and we will continue to keep William's family informed as our investigation progresses."
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