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.
She was the country's longest-serving monarch after reigning for 70 years.
An email from Renfrewshire Council's head of corporate governance to the group leaders said the national guidance was that the day-to-day functions of local authorities should continue as normal.
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However, the communication said members could move upon commencement of scheduled meetings that they be adjourned until a later date. That would become the outcome if it was the majority view of the board.
In a statement, Councillor Graham said: "Our group thinks it is inappropriate for normal council business to continue.
"We hope the convener of the Leadership Board postpones this meeting until after the funeral of Queen Elizabeth II, in line with the Scottish Parliament and many other councils across Scotland."
Councillor Graham confirmed his group's stance in a response to the head of corporate governance and other group leaders, adding that he "really hoped" the Leadership Board could be rescheduled.
However, he felt Thursday's Regulatory Functions Board should go ahead, describing it "non-political".
A council spokesperson said in a statement: "Elected members can consider adjourning the Leadership Board at the start of the meeting.
"The meeting can also be cancelled if all of the political parties represented on the board put in a written request in advance."
Councillors are due to consider an extensive package of support for struggling residents – including winter clothing payments and plans to introduce warm spaces – at the Leadership Board.
The proposals have been drafted as rising inflation and energy costs continue to inflict several financial pressure on people in the area.
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