It has already agreed in principle to support the move by the Queens Trust.
The group needs the local authority to use its shareholding in the Dumfries club to get the issue discussed at its annual general meeting.
The supporter organisation hopes to secure a director's role or a place as a non-voting member of the board at the Scottish Championship side.
The Dumfries common good sub-committee is being asked to back a resolution being taken to the club's AGM.
The club has said that its door is "always open" for an "open and honest dialogue" with fans.
It has also appointed a supporter liaison officer to improve the relationship between fans and directors.
Dumfries and Galloway Council is being asked to back a fans' bid to get a seat on the board at Queen of the South.