The plea comes just ahead of a potentially key meeting between the two sides on Monday. So far, the Irish border backstop issue has proved easily the single biggest obstacle to a deal being struck between the EU and UK.
Boris Johnson will meet European Commission President Jean-Claude Juncker, his first with the outgoing EU leader, later on Monday.
Ahead of the meeting, Irish MEP Billy Kelleher (Renew Europe/Fianna Fáil) told this website that any proposals that "ensures frictionless travel and trade on the island of Ireland are to be welcomed."
He added, “Ensuring that farmers, businesses and ordinary people can go about their daily business is a priority for me and my party."
“I hope that the UK Government actively considers this proposal and steps back from the brink in terms of a No Deal Brexit. The outcomes of which would be both calamitous and catastrophic for the people of the island of Ireland,” said Kelleher.
Further comment came from another Irish MEP, Martina Anderson, of the GUE/Nordic Green Left, who told The Parliament Magazine, “I have very little faith in the British government coming up with any workable plan.”
The Sinn Fein deputy added, “They are seeking to wriggle out of their responsibilities under the Good Friday Agreement, as they are seeking to wriggle out of their commitments in the Joint Report of December 2017.
“I hope that the UK Government actively considers this proposal and steps back from the brink in terms of a No Deal Brexit. The outcomes of which would be both calamitous and catastrophic for the people of the island of Ireland" Billy Kelleher MEP
“I don’t believe that any plan they come up with will genuinely uphold the Good Friday Agreement or genuinely protect the all-island economy and north-south cooperation. The backstop for the north of Ireland must be maintained in the Withdrawal Agreement."
"In the context of Brexit, a border in the Irish Sea would make sense, with the North remaining in the single market and customs union. But I very much doubt that the British government will propose that but it’s the only solution to the Brexit mess foisted upon the people of the north of Ireland who voted to remain."
She added, "Of course, the real solution would be a referendum on Irish reunification, as provided for in the Good Friday Agreement and legislated for in the 1998 Act."
The EU's chief Brexit negotiator Michel Barnier will also attend the meeting in Luxembourg, while Boris Johnson will be accompanied by UK Brexit Secretary Steve Barclay and Downing Street's Brexit representative David Frost.
Last week UK MPs passed a law that will require Boris Johnson to ask the EU to extend the 31 October deadline if a deal was not agreed by 19 October.
Johnson told the Mail on Sunday newspaper in the UK that, on that date, the UK would break out of its "manacles" like cartoon character The Incredible Hulk - with or without a deal.
The UK side says it has presented ideas on customs and manufactured goods, while there has been further discussion on the non-binding political declaration which sits alongside the Withdrawal Agreement and outlines the future relationship between the two sides.
However, senior EU figures including David Sassoli, Parliament’s new President, contradict such claims.
“I have very little faith in the British government coming up with any workable plan. They are seeking to wriggle out of their responsibilities under the Good Friday Agreement, as they are seeking to wriggle out of their commitments in the Joint Report of December 2017" Martina Anderson MEP
Speaking last week, the Italian MEP insisted that talks are at a standstill and the current suspension for five weeks of the UK parliament has made a no-deal Brexit on 31 October more likely.
Barnier has updated members of the European Parliament and said "we do not have reasons to be optimistic...the ball is clearly in the British court."
An EU source said, "We want to keep this going but at some point the UK needs to give us a proposal. We can't negotiate without one."
The idea of an all-Ireland zone for food and animals (in which the Republic of Ireland and Northern Ireland would follow the same rules after Brexit) has been explored and officials say the UK has presented "preliminary ideas" on how any solution in Ireland could involve the consent of all parties.