It was claimed in the Dáil this week that there's "no appetite" in Offaly to achieve global recognition.
There are several concerns locally about designating Clonmacnoise as a UNESCO World Heritage Site.
The slow pace of getting approval for Irish monastic sites was raised in the Dáil earlier this week.
The proposal for Clomacnoise was first mooted in 2009, but a lack of support meant it failed to materialise.
But Fianna Fáil councillor Eamon Dooley says getting UNESCO status for Clonmacnoise would require moving the local cemetery and imposing a large exclusion buffer zone.
He says plans such as this need buy-in from residents: