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".

Two new pictures of the third in line to the throne have been released to mark the occasion on Tuesday.
One shows Prince George transfixed by a butterfly, while a second shows him with his mother and father, staring at an object above their heads.
In a statement the duke and duchess thanked people for their "warm and generous good wishes".
The couple said: "We would like to take this opportunity on George's first birthday to thank everyone over the last year, wherever we have met them, both at home and overseas, for their warm and generous good wishes to George and our family."
The duke and duchess are expected to hold a birthday party for the young prince on Tuesday at their Kensington Palace apartment.
Prince George was first revealed to the public on the steps of St Mary's hospital in London on 23 July 2013.
The couple and their son were also warmly welcomed when they travelled to New Zealand and Australia on a royal visit.
An image of Prince George was released on Sunday and the new pictures published on Monday night complete the set.
One shows the young royal fascinated by a blue morpho butterfly, which has landed on his father's hand.
The other image shows the two future kings and the Duchess of Cambridge looking at an object above their heads.
The pictures were taken a few weeks ago at London's Natural History Museum by the Press Association's John Stillwell.
Just like many of the other 2,200 babies born in the UK on the same day, last year, George Alexander Louis will mark his birthday with a party.
His though, will be at a palace and his guest of honour will be his great-grandmother, the Queen.
Prince George offers the institution she has served for decades the prospect of renewal, but not - as things stand - for many years to come, after the reigns of his grandfather and his father.
For now, the emphasis will be on him leading as normal a life as possible - not easy when you're a prince growing up in a palace - and on shielding the child, destined to be a king, from too much public exposure.
But George will continue to fascinate and he will continue to be showered with presents.  A Kenyan women's group has given the baby prince, who will one day inherit the Duchy of Cornwall's 53,000 hectares, two fattened bulls, a heifer and a goat.
Sir Michael Dixon, director of the Natural History Museum, said: "It was a real pleasure to host our patron and her family earlier this month. They seemed to enjoy their visit, wandering through the tropical plants as brightly coloured butterflies took flight around them."
The duchess became patron of the museum in April last year.
Some of north Wales' most important heritage sites could close if no-one is found to take them over, Anglesey council has warned.
[NEXT_CONCEPT]
The Duke and Duchess of Cambridge have thanked well-wishers as Prince George celebrates his first birthday.