The group wants Gen Prem to act as a ''middle man'' to work under His Majesty the King in finding ways to end the country's turmoil.
Group member Gen Saiyud Kerdphol and chairman of the People's Network for Elections said Gen Prem should function as an intermediary between the King and other parties, since His Majesty has said he cannot be involved directly with political problems.
Gen Saiyud said Gen Prem could consult judicial agencies, the military and leaders of other organisations to draft a proposal to end the crisis. That proposal could then be forwarded to the King, who could instruct the statesman on how to proceed.
He argued the statesman's post was originally created to carry out royal commands in times of crisis and perform responsibilities as directed by the King.
Consultation on a draft proposal to fix the country's problems should start immediately, Gen Saiyud said, adding: ''We do not have to wait for the country to go into a leadership vacuum first.''
He was referring to the possibility of caretaker Prime Minister Yingluck Shinawatra and her cabinet being removed should the Constitutional Court rule that she violated the charter when she transferred National Security Council chief Thawil Pliensri in 2011.
Another member of the group, Pramote Nakhonthap, said the King's wishes should be heeded in the event of national emergencies. But he insisted the group is not asking the King to install a prime minister.
Mr Pramote said no statesman had assumed such a ''middle man'' role before, but said that does not mean the King could not choose to exercise his power that way.
He downplayed concerns that inviting Gen Prem to mediate could provoke red-shirt supporters who are openly critical of the statesman. He said critics should not be allowed to prevent people from doing what is right.
As citizens, the group had the right to urge Gen Prem to help, which is perfectly constitutional, Mr Pramote said.
Meanwhile, religious authorities are holding a nationwide mass prayer for national peace on April 18.
The event — involving the Sangha Council, the Office of the Chularatchamontri, the Catholic Bishops' Conference of Thailand, the Evangelical Fellowship of Thailand and groups representing Brahmin-Hindu and Sikh religions — will be held at designated venues for each religion.