In April, the authority launched a search for organisations to run the sites, which include Wales' only working windmill and a Victorian prison.
The council said budget cuts had forced it to give up the attractions.
Only one organisation has so far expressed interest.
Beaumaris courthouse and gaol, and Melin Llynnon and its ancient roundhouses, near Llanddeusant, are currently run by the council's museums and culture service.
Anglesey's head of learning, Delyth Molyneux, said: "Given the significant financial pressures faced and the fact that these sites don't form part of a statutory service, there's a real possibility that they will have to be mothballed or closed altogether if no one else can be found to run them."
Politicians had expressed concerns the courthouse and gaol could be sold to developers.
Beaumaris town councillor Jason Zalot said a sub-committee had been set up to look into running the Beaumaris gaol and courthouse.
"It's a massive undertaking for us. But bear in mind the town council have already taken over the management of the toilets.
"We have proven that we can work hand in the hand with the council."
Llynnon is currently closed and both Beaumaris courthouse and goal are open on weekends and during upcoming half term holiday.
Council leader Ieuan Williams said the council would do its best to support "interested parties" in order to ensure a smooth handover "as the success of these attractions is very important to us".
Some of north Wales' most important heritage sites could close if no-one is found to take them over, Anglesey council has warned.