The former Scotland manager made a surprise return to his old club this week on a three-month consultancy basis but will continue to serve as technical director at Dens Park.
However, the move would appear to go against rules regarding persons having involvement with two clubs at the same time.
According to the Daily Mail, neither club is thought to have run the appointment past the SFA.
A Celtic statement read: “Gordon will spend his time looking at our activities within our football academy, our women’s team and newly established B team, with a view to providing recommendations to chief executive Dom McKay on how we can develop these areas further, feeding into wider work ongoing across the club.”
Strachan’s comments – mainly focusing on his delight at returning to the club and looking forward to working with McKay – ended: “I also look forward to continuing my work at Dundee where I have established a great working relationship and close bond with the manager, managing director, staff and coaches.”
The Dark Blues, who were beaten 6-0 by Celtic at the weekend, were then forced to issue a statement clarifying that Strachan would indeed be continuing in his role at Dens Park.
"This evening it has been announced that Gordon has taken a short-term consultancy role with Celtic but this will not affect his role with Dundee. We have been in constant communication with Gordon and Celtic throughout this process,” a brief statement confirmed.
Article 13 of the Scottish FA’s regulations states: “Any person who (i) is a member of a club, (ii) is involved in any capacity whatsoever in the management or administration of a club or (iii) has any power whatsoever to influence the management or administration of a club may not take up any such role with another club until such time as the Scottish FA is reasonably satisfied that such person has ceased to hold such role in the first club” without written consent.
The SFA is due to discuss the unconventional arrangement at the next meeting of the Professional Game Board, expected to convene next week.