Shrews revealed last week that chief executive Brian Caldwell had left the club and, in a statement, Town announced they were creating a new football management structure.
Chairman Roland Wycherley said in an open letter to fans chief executive of the foundation, Jamie Edwards will be part of the new hierarchy – although his job title is not yet known.
As things stand, the other confirmed members of the football board are Peter Brophy as chief operating officer and Duncan Montgomery as finance director.
But arguably the most important appointment of them all is on the football side of the board and Moore met with the club on Tuesday after an initial approach over the weekend.
The 49-year-old joined the Robins in August 2018 and has been of interest to other League One clubs in the past with Forest Green Rovers keen to land the former and Solihull Moors and Barrow man.
Speaking last week, Cheltenham chairman David Bloxham said: "Shrewsbury have asked for permission to speak to Micky and as a matter of courtesy and principle we always tell our staff when approaches are made.
"We've told Micky we don't want him to leave but have given him permission to speak with Shrewsbury."
It does little to clear up the uncertainty around manager Steve Cotterill's future at the club – with it still unclear how he fits into the new management structure.