The Taoiseach has said the Government needs to "step up our preparations for a no-deal scenario", although he thought that outcome is unlikely.
Last week an internal EU document seen by RTÉ News warned all member states, companies and stakeholders to step up preparation for such a situation.
Leo Varadkar said last night’s vote in the House of Commons to amend Britain’s Custom Bill should not give any cause for panic about the backstop ensuring that there will be no hard border on the island of Ireland.
The British government has already proposed a temporary "backstop" arrangement to prevent a hard border on the island of Ireland until December 2021.
Mr Varadkar said the vote "shouldn't give us any reason to change our position" in Brexit negotiations.
He acknowledged there is a "lot of political instability in London and turmoil in Westminster, and we're going to see a lot more twists and turns in the months ahead."
The Taoiseach added that he believed a Withdrawal Agreement could be reached by October, but said that with the current instability in Britain that "we can't make assumptions that the Withdrawal Agreement will get through Westminster.
"It's not evident, or not obvious, that the govt of Britain has the majority for any form of Brexit quite frankly," he said.
Earlier the Department of Foreign Affairs said it has been the UK government's position that customs checks between Northern Ireland and the rest of the UK are are unacceptable, so it is not surprising that the amendment to the Customs Bill was accepted by the Commons.
In a statement, the department said: "the UK has also repeatedly committed to avoiding a hard border the island of Ireland, most recently in last week's White Paper. In fact the Withdrawal Act, passed last month, legally binds the UK government to this commitment.
"The Irish Government has always been clear that our first preference, in terms of avoiding a hard border, is through the closest possible future EU-UK relationship.
"It is not for the Irish Government to comment on the internal politics and legislative processes of the UK…What matters is that the British government is able to engage in serious negotiations with the Commission.
"…It is fully accepted and understood that there can be no Withdrawal Agreement without a legally operable backstop ensuring that there will be no hard border on the island of Ireland".
Fianna Fáil's spokesperson on Brexit said she believed Ireland had to be very worried after the Customs Bill vote.
Speaking on RTÉ's Morning Ireland, Lisa Chambers said she was not sure "where we go from here".
Ms Chambers said the most worrying amendment that was passed was the one that related to the backstop.
She said in her opinion yesterday's vote in the Commons had essentially made a backstop unlawful and therefore it was not possible to contain it in the Withdrawal Treaty.
She said: "The EU has reiterated its point several times that there will be no withdrawal treaty unless the UK adheres to its commitments made last December and last March that it would agree to a legally binding backstop, which now in my view is now not possible.
"This is the very first time we've had something passed into law in the UK which does effectively preclude the backstop.
Labour Leader, Brendan Howlin has said he believes that the best thing for Ireland now is for a change of government in Britain and for the Labour party there to be in office.
He told RTÉ’s 6.1 News that the official position of Britain's Labour party is to stay within the Customs Union and the Single Market for at least a four year transition period.
He said this would allow a period of time to work out an arrangement between the European Union and the UK that would be in Ireland's interest.
Deputy Howlin said that Ireland needs to ensure that the backstop is meaningful by insisting on clarity in advance of October and he believes that the issue should be finally resolved at the special council meeting in September which has been called to discuss migration.
Meanwhile, European Parliament Vice President Mairead McGuinness has said that in her opinion the backstop agreement and the Chequers document were "not dead in the water" and that it was very dangerous talk to say otherwise.
Ms McGuinness said it had already been agreed that there would be no hard border on the island of Ireland.
She said as far as she was concerned, the EU was working off a White Paper that it received from the British government, which will be used to work out a framework for the future relationship between the UK and the EU.
However, she said the events of last night had thrown some "chaos" into the situation.
The UK's International Trade Secretary Liam Fox has also insisted that the Chequers cabinet compromise on Brexit was not dead.
He told BBC Radio 4's Today programme: "The wording of the amendment yesterday was very close to the wording in the government's White Paper.
"It looked in fact as a bit of a cut and paste from the White Paper."
Dr Fox said he wanted to see a "people's Brexit", stating: "We can't please everybody.
"We have to have a compromise position that enables the country to get an agreement with the European Union.
"Here in Britain there is far too much negative, self-doubting pessimism in this process."
Defending moves to cut the parliamentary session short, Dr Fox said: "Well, of course, you have to draw the distinction between parliament and government, because government doesn't stop over the recess."
On calls for a new referendum, Dr Fox said: "What if we have a referendum and it goes the other way? Do we have best of three?"