It follows the annual general meeting of the DG Unlimited group this week.
Cathy Agnew, who chaired the meeting, confirmed that a "data protection issue" had been raised during proceedings.
The council said it was providing help to investigate the matter.
DG Unlimited was set up after a previous region-wide arts organisation, dgArts, ceased trading.
It operates four arts hubs across the area covering Annandale and Eskdale, Nithsdale, the Stewartry and Wigtownshire.
The investigation is understood to centre on members' private details being shared without the proper consent.
Ms Agnew told the BBC Scotland news website: "There was an issue raised by two members of DG Unlimited.
"There is an issue about data protection."
Dumfries and Galloway Council confirmed it had been made aware of a "potential data protection issue".
"As it is an independent organisation, this isn't a council matter," a spokesman said.
However, he said council officers were providing "advice and assistance".
Dumfries and Galloway Council has been asked to help investigate allegations of a data protection breach at a regional arts organisation.