Commending the work behind the scenes to organise the weekend's successful open day at the Drumclay Crannog, councillors were in agreement at their monthly meeting that the precious items should be retained in their home County.
Over 700 people flocked to the crannog on Saturday to see for themselves the site 'international significance'.
Guided tours and introductory talks at Fermanagh County Museum were provided.
Sinn Fein councillor Tommy Maguire said the open day had allowed the opportunity to demonstrate Enniskillen's "sophisticated culture" from long ago.
"The depth of the history involved here is very exciting," he said.
Fellow party member, Brian McCaffrey said Environment Minister Alex Attwood was keen to hold another open day in the new year, such was the level of interest in the crannog.
Acknowledging that the site and its contents "tell the story of our County's past", Fermanagh District Council chairman, Thomas O'Reilly said the items should remain here.
"We would like to see that these artefacts are retained in the County and have them in our museum. Some times these things disappear by default, but this is an ideal opportunity to tell some of the history of the County."
Chief Executive Brendan Hegarty said he had personally raised the matter with Mr Attwood.
"He is well aware that's our view," Mr Hegarty assured, "Subject to funding down the line we are looking to enhance the Enniskillen Museum and we feel it is only right that these artefacts should be retained in Fermanagh. Therefore we will be making the request."
Independent councillor Bernice Swift agreed that the Council should "make the demand that the items be stored here for future generations".
"It is precious that we have this fantastic crannog here in Fermanagh," she said, "I look forward to our history being rewritten.
"I wonder has anyone any concerns now about the road being delayed. Is it possible that they (Roads Service) can just rail road over this precious site? I for one wouldn't want to see this happening until the archaeologists are satisfied their work is complete."
UUP councillor Basil Johnston queried whether the Northern Ireland Environment Agency had known a crannog existed at the site when Roads Service planning approval was given to commence work on the Link Road.
"We can't speak for them but we believe that they did know," said Mr Hegarty.
The Chief Executive said excavation had been allowed until the end of December.
"It is our view that the archaeologists don't think work will be completed by the end of December," he added.
Director of Leisure, Development and Arts, Robert Gibson, explained the work involved the exploration of some five metres.
To date, only three metres had been explored.
"We have waited long enough for roads in Fermanagh, I'm happy to wait until we are satisfied the work is completed," said Ms Swift.