Update 9.15pm: Leo Varadkar says a US-Canada-style border would not be an acceptable option for the Irish border post-Brexit.
Theresa May has indicated she is looking at the customs arrangements between the US and Canada as a way of solving the Irish border issue created by Brexit.
While Mr Varadkar said he had not heard the UK Prime Minister's comments at Westminster so he did not want to comment directly, he did discuss his trip to the US-Canada border.
"I visited the US-Canada border, I visited it back in August, and I saw a hard border with physical infrastructure, with customs posts, people in uniforms with arms and dogs and that is definitely not a solution that is one that we can possibly entertain," Mr Varadkar said.
The Taoiseach made the comments at Government Buildings in Dublin after a meeting with Luxembourg Prime Minister Xavier Bettel.
"That is definitely not a solution that we can possibly entertain" – Taoiseach rules out US/Canada style border pic.twitter.com/LA0mSdVfYQ
— RTÉ News (@rtenews) March 5, 2018
Mr Bettel said after the terrorist attacks in Paris in November 2015 Luxembourg had border controls and they had impacted the economy.
"For the moment I can't tell you about Theresa May's plans," Mr Bettel said.
"I want to see them, I want to read them, and then I'll be able to tell you if these are acceptable or not."
Asked if he was frustrated by the UK Government, Mr Varadkar said: "No, no, I never get frustrated by the British and I'm always very patient."
Mr Bettel said solutions were needed because the time is ticking.
"We started with a hard Brexit, now some people spoke about a soft Brexit, I want an intelligent Brexit," the Luxembourg PM said.
"We should be able to find the best way where we don't punish but it's not time for cherry-picking and we need a level playing field."
Update 8:18pm: Theresa May has indicated she is looking at the customs arrangements between the US and Canada as a way of solving the Irish border issue created by Brexit.
But the UK Prime Minister, who has insisted she remains committed to avoiding a hard border between Northern Ireland and the Republic, was mocked over the comparison - with one MP telling her the US-Canada frontier has armed guards.
The issue of the border is one of the most complex problems to be resolved in the Brexit process and Mrs May said the British Government was examining examples from around the world.
Updating MPs following her Mansion House speech on Brexit Mrs May said: "There are many examples of different arrangements for customs around the rest of the world.
"Indeed we are looking at those, including for example the border between the United States and Canada."
Brexit Secretary David Davis has previously highlighted the operation of the frontier near the Ford plant in Detroit as an example of how cross-border trade can operate smoothly.
But when it was later pointed out to the Prime Minister by shadow Brexit minister Jenny Chapman that there were "armed customs guards" at the US-Canada border, Mrs May said the Government was looking at arrangements "in a number of countries".
Labour MP Chris Leslie, a supporter of the pro-EU Open Britain group, said: "The Government has repeatedly ruled out any return to a hard border on the island of Ireland, as well as any new physical infrastructure at the border.
"But anyone who's visited the US-Canada border will know that it's a very hard border indeed, with customs checks, barriers and armed guards.
"Leave campaigners told us there'd be 'no change' to the border if we voted to leave the EU. That promise is looking less deliverable by the day."
In the Commons, Mrs May received support from Tories on both sides of the Brexit divide, significantly including potential rebels who could inflict defeat on the Government over the plans to leave the EU's customs union.
Leading rebel Anna Soubry said no-one could doubt Mrs May's "determination" to get the "very best deal for our country in these most difficult of negotiations" but warned there would be "considerable administrative costs".
Among the prominent Brexit-backers, former Tory leader Iain Duncan Smith urged the Prime Minister to send a message to Brussels: "When she gets into negotiations with her European counterparts about trade arrangements, could she remind them that cake exists to be eaten and cherries exist to be picked?"
Meanwhile Taoiseach Leo Varadkar ruled out formal three-way talks between the UK, Ireland and the EU to look at Mrs May's Brexit offer.
Updating MPs on her plans, Mrs May said the UK and Irish governments and the European Commission "will be working together" to avoid a hard border on the island of Ireland.
But Mr Varadkar said it was not in Ireland's interests to take part in formal three-way talks regarding the wider Brexit issues, and that what was needed was for Downing Street to produce more detailed proposals.
My impressions of the Canada-US border when I visited it last year. https://t.co/UUBNCxdEpG— Leo Varadkar (@LeoVaradkar) March 5, 2018
Mrs May again told MPs there would be no return to a hard border between Northern Ireland and the Republic.
She has rejected Brussels proposals that would see Northern Ireland kept in an effective customs union with the EU as a fallback in case other solutions cannot be found.
"As Prime Minister I am not going to let our departure from the EU do anything to set back the historic progress made in Northern Ireland," she said. "Nor will I allow anything that would damage the integrity of our precious union.
"The UK and Irish governments and the European Commission will be working together to ensure that we fulfil these commitments."
Before Mrs May's statement, the Varadkar said: "There won't be tripartite or three-way talks.
"What will happen is that there will be talks between the EU 27 and the UK, and Ireland is part of the EU 27 and we're much stronger by the way as one of 27."
Mr Varadkar added consultations could take place between the two governments about issues that are unique to Ireland.
"We will of course have negotiations about what could be done to avoid a hard border, but what we won't be getting into is a negotiation with the UK, or a three-way negotiation," Mr Varadkar said.
"That's not in our interest and not the way that this can be concluded."
The Taoiseach told RTE's Morning Ireland programme that he gave Mrs May's Brexit speech on Friday a guarded welcome, but that detail "written down in black and white" was now needed from the UK Government.
Amid the ongoing row over the border, Sinn Féin leaders Mary Lou McDonald and Michelle O'Neill travelled to Brussels to meet EU chief negotiator Michel Barnier.
Mrs McDonald said: "The Taoiseach is absolutely correct to say that the substantive negotiation is between the British government on one hand and the EU member states collectively on the other and in our view it is important to maintain that dynamic."
Mr Barnier, who is due to meet DUP leader Arlene Foster and deputy leader Nigel Dodds in Brussels on Tuesday, said his discussions with Sinn Fein had been positive.
He tweeted: "Essential to listen to all voices in Northern Ireland."
Good meeting this morning with @MaryLouMcDonald and @moneillsf @sinnfeinireland - essential to listen to all voices in Northern Ireland. Meeting @DUPleader tomorrow. Working to ensure the #UK’s orderly withdrawal #Brexit pic.twitter.com/ckNDwO1eeS— Michel Barnier (@MichelBarnier) March 5, 2018