The European Commission has asked for details from the football club and Glasgow City Council after receiving a number of complaints about the purchase of land around Celtic Park.
The Commission has not opened a formal investigation but is analysing information.
The Scottish champions said purchases had been made on "commercial terms at market rates" and it will assist the Commission fully.
In addition, the club's Lennoxtown training centre was built on the grounds of the former Lennox Castle Hospital in East Dunbartonshire, which was formally owned by NHS Greater Glasgow and Clyde.
This is not part of the Commission's analysis.
However, the Scottish Government said it was aware of the development and is discussing it with the health board and the club.
A Glasgow City Council spokesman said: "A complaint was made to the European Commission on historic land deals around Celtic Park.
"The Commission is legally obliged to investigate all such allegations, and the council was happy to provide information on these transactions."
A statement on Celtic's website read: "Celtic Football Club operates to the highest standards and with the utmost integrity.
"At a time when the Club is committed to investing in and improving areas around Celtic Park, not only for Celtic supporters but for the benefit of the local community, it is sad that these baseless accusations have been raised with the European Commission.
"Any suggestion that Celtic has been the beneficiary of state aid is preposterous - as ludicrous as any suggestion that we have benefited from soft loans from our bankers.
"The historic transactions referred to were negotiated with the Council on commercial terms at market rates.
"The Club will assist the Commission fully with the process and will not be deterred from our work to improve our local area."
A Scottish Government spokesman said: "We are aware of these allegations and we are working with the relevant parties to help the Commission to investigate this case consistent with our role to ensure public funds in Scotland are used in compliance with EU state aid regulations.
"The transaction between the Health Board and Celtic is not being investigated by the European Commission but we are also aware of it and are discussing it with the relevant parties."