Conservative councillors believe it would be "inappropriate" for a key council summit to go ahead just under a week after the passing of Queen Elizabeth II.
Councillor Neill Graham, the party's group leader, intends to move for the Leadership Board to be adjourned on Wednesday and has confirmed that if his peers vote for the meeting to go ahead, he and his depute will opt to leave.
A period of national mourning was announced on Friday, the day after the Queen's death at the age of 96, and will continue until the end of the day of her State Funeral on Monday, September 19.
