"The Taoiseach made it clear in the interview that the Government is determined to avoid a no-deal scenario and the consequent risk of a hard border", the spokesman said.
But the Irish government and rest of the EU's member states have insisted the deal is not up for renegotiation.
"We're 60 days out from the reality of Brexit with or without a deal", said Border Communities Against Brexit organiser Tom Murray.
He told BBC One's Andrew Marr Show: "The European Parliament will not ratify a Withdrawal Agreement that doesn't have a backstop in it".
Varadkar, too, despite his incendiary rhetoric on a militarised border, is said to be privately briefing Irish politicians that customs checks will be carried out at European Union ports such as Calais rather than the border with Brexit Britain - on which the Irish economy is massively dependent in terms of trade.
The demonstration by Border Communities Against Brexit took place on the old Dublin road near Carrickcarnon this afternoon.
After a rally calling on Westminster to avoid the return of a hard border along the 500 kilometre boundary, locals smashed down the mock wall using sledgehammers in a jubilant atmosphere.
We want to help Mrs May but we need a stronger indication from your MPs about where we go from here and what they want.
He said Brexit discussions had been dominated by speculation on its potential impact on trade and the economy, but that for Ireland it was about ensuring continued peace on the island and that the progress made in 20 years since the Good Friday Agreement was not lost.
Mr Czaputowicz told reporters in Brussels on Monday: "I've just discussed that idea with my Irish counterpart Simon Coveney and also with British foreign Secretary Jeremy Hunt today, I think it would be one of the solutions".
Irish taoiseach (prime minister) Leo Varadkar has warned earlier in the week that a No Deal Brexit could "involve people in uniform... possibly a police presence, or an army presence to back it up" - but backpedalled rapidly after his rhetoric as "reckless and irresponsible" by the self-same Mary Lou McDonald mentioned above and opposition politicians in his own country.
The Sunday Times newspaper raised eyebrows by reporting that officials are looking at what powers they might need in the event of any civil disorder, including imposing martial law.
DUP deputy leader Nigel Dodds said the Taoiseach should take steps to quell border fears. "We could work with a Canada-model with special arrangements for Northern Ireland", Mr Varadkar said.
Speaking from Davos where he is attending the World Economic Forum, Mr Varadkar said it was "very unlikely" that Brexit would not happen at all.
These are in answer to critics who dislike the backstop because they believe it keeps the United Kingdom too closely aligned to the European Union and fear that it could become permanent.