Council leader Cammy Day revealed the idea was being explored for council-owned attractions, venues and libraries to help provide families with places where they can "feel safe, warm and at ease”.
He said the council was looking at creating the temporary spaces so people could also access “support and advice to help alleviate food and fuel poverty".
It comes after councillors in Glasgow unanimously voted last month to establish warm banks in the city this winter.
Cllr Day pledged his council’s help for people facing "the toughest financial squeeze of their lifetimes”, with many concerned about their energy costs and heating their homes “or, worse still, losing them”.
The council leader urged property owners affected by new rules and legislation designed to reduce the number of buildings used for short-term letting to play their bit in helping.
Museums and galleries run by the council include the City Art Centre, the Museum of Edinburgh, Lauriston Castle, the People’s Story Museum and the Museum of Childhood.
Other venues include the Usher Hall, the Assembly Rooms, the City Chambers, the Festival Theatre and Churchill Theatre.
Cllr Day said: “The true scale and difficulty of the cost-of-living is becoming ever clearer.
“Let’s be in no doubt – this is a huge debt crisis, with people facing the toughest financial squeeze of their lifetimes, and we, and our partners, are working extremely hard to lessen the impact of the crisis and make a real difference.
“I am conscious that many of our residents will rightly be concerned about energy costs and heating their homes – or, worse still, losing them.
"One option we’re considering is the provision of ‘warm and welcoming spaces’ across the city.
"Council venues such as libraries, museums and galleries could offer free activities for a variety of age groups, providing spaces where people can feel safe, warm and at ease. These spaces would also be a place where residents can access support and advice to help alleviate food and fuel poverty.”
Cllr Day added: “With pressure on our housing stock greater than ever, I was delighted to see Edinburgh becoming Scotland’s first short-term let control area last month and I hope to see property owners do the right thing and turn unsuitable holiday lets back into homes.
"Landlords can also sign over to private sector leasing where they will receive market rates and guaranteed tenancies through the council."
SNP councillor Euan Hyslop: said: "It's devastating we are in this position, discussing emergency measures to mitigate the risk to life of this energy crisis in an energy rich country.
“With winter approaching, bills rising rapidly, and a looming risk of gas shortages, the impact of this crisis is still to be seen. In light of this, it is right that the council explore all options for ‘warm spaces'.
"This should include freeing up space within museums and art galleries that will be open to the public and will have the heating on anyway.
"The use of these cultural centres will also allow those in need to overcome the fear of stigmatisation associated with asking for help that we see at food banks."