The Labour leader Brendan Howlin has claimed a number of European member states are considering trying to kick the issue of the Irish border down the road.
It is the key stumbling block that is preventing Brexit talks between the EU and the UK from progressing.
President of the European Commission, Jean-Claude Juncker, has said it would be a catastrophe if there was no deal on Brexit.
Brexit talks are stalled until tomorrow's EU Council meeting.
The proposed backstop, the deal that would prevent a hard border with Northern Ireland if no new trade arrangement can be reached, is proving contentious.
Mr Howlin has said he is now hearing reports other European countries want to kick to down the road.
He said: "I now hear for the first time that there are actually suggestions being made that the Irish backstop issue should be postponed.
"I think that is very dangerous for us and I think it's something we have to resist."
Fianna Fáil's Brexit spokesperson, Lisa Chambers, has said there are lots of rumours flying around when it comes to Northern Ireland, but she maintained the backstop has to be dealt with at this stage of the talks.
Ms Chambers said: "It is my view, and the view of the Fianna Fáil party, that if we allow that issue to get kicked down the road, we could find ourselves in a very lonely situation
Tánaiste Simon Coveney is due to meet the EU's Chief Negotiator Michel Barnier in Luxembourg before the Taoiseach Leo Varadkar travels to Brussels tomorrow for the EU Council's Brexit summit.
