The EU's chief Brexit negotiator was giving his first substantial reaction to the U.K.'s alternative "backstop" proposal to avoid a hard border in Northern Ireland if negotiators fail to reach agreement.
"Our backstop can not be extended to the whole U.K.", Barnier said.
Others questioned the substance of the proposal.
The Best for Britain called on the country's Prime Minister Theresa May to present her clear proposals in autumn on the country's future after the withdrawal, as it was unacceptable "to hold the country hostage" of internal governmental divisions.
The issue continues to be a thorn in the side of Prime Minister Theresa May, as debate rumbles on about how best to avoid a hard border between the two Irelands. It would have signed us up to the rules of the EU Customs Union (a central pillar of Brussels rule) if it was ever activated.
"Does the temporary nature of the customs arrangement mean that several adaptations will now be needed?"
The backstop solution at the heart of the Brexit talks is due to snap into place to avoid a hard Irish border should an adequate free trade deal or technological answer not be in place by the end of the transition period at the end of 2020. "How does that fit with the requirements of our Value-Added Tax system?"
If backed by a majority of MPs, the proposed amendment would commit the Government to holding a vote before the formal date of Brexit on March 29 2019, with the UK's withdrawal automatically halted if voters reject Mrs May's plan.
He said some progress had been made in talks and that a June 28 summit on Brexit could see a breakthrough.
"I've always been clear that this was time-limited and as we've said today [Thursday] the end state at the latest will be in by the end of 2021 and we will be working to ensure that it is in earlier than that".
A campaign for a another European Union referendum - this time on the final Brexit deal - has been launched in London.
The debate will test May's ability to broker a compromise with those in her Conservative Party who, like many members of the upper house, want to keep a relatively close relationship with the European Union after Brexit. He argued that such solution raises questions of an à la carte access to the single market, services, goods or people that is unacceptable to the EU.
"There's a big prize for the United Kingdom here at the end of this and that prize is being able to be in control of our money, our laws and our borders, being more secure and more prosperous and being able to make those free trade deals with the rest of the world and a bright future".
In the 2016 referendum, Northern Ireland voted 56 percent to remain in the European Union but, like Scotland, was outvoted by England and Wales and the overall result was 52 percent for Brexit.