Ms Truss announced her decision on Thursday outside 10 Downing Street after just 44 days in office, making her the shortest serving Prime Minister in British history.
It comes after her meeting with the chairman of the 1922 committee, Sir Graham Brady. Both agreed that a leadership election will be completed within the next week and Ms Truss will remain as PM until her successor has been chosen.
After publicly backing her “vision to move the country forward” councillor Thomas Kerr believes Ms Truss had no other option available to her.
He said: “Liz Truss has made a tough choice but ultimately it is the right one. There was no other decision that could have been made after recent events.
“I wish her and her family well going forward.”
In July councillor Kerr said he had been unsure of who Boris Johnson’s successor should be but felt Ms Truss could help rebuild the economy and help families suffering most from the cost of living crisis.
He now says his party must unite and has his own ideas on who the next Prime Minister should be.
Councillor Kerr added: “What is crucial now is that the Conservative and Unionist Party unites and moves quickly and elects a new Leader and Prime Minister to get on with tackling the huge challenges facing the people of Glasgow and the whole of the United Kingdom.
“For me Penny Mordaunt offers our country the stability and unity we need and I hope she throws her hat into the ring.”
A new leader is expected to be chosen by next Friday.