Irish state broadcaster RTE reported that the suggestion sent to the EU by the UK would lead to customs posts being built at the edge of a ‘clearance area’ along the border.
Deputy Irish premier Simon Coveney dismissed the plans as a “non starter”, saying Northern Ireland and the Republic of Ireland "deserve better".
But in interviews at the Conservative Party conference in Manchester this morning, the Prime Minister insisted that the reports were “not what we’re proposing at all”.
He said a “good offer” would be made and claimed critics were seeking to “needlessly distort” the UK Government’s plans.
Speaking to the BBC’s Breakfast programme, Mr Johnson said: "They are not talking about the proposals we are going to be tabling, they are talking about stuff that went in previously.
"But clearly this is the moment when the rubber hits the road.
"This is when the hard yards really are in the course of the negotiations.
"The difficulty really is going to be around the customs union and to what extent Northern Ireland can be retained within EU bodies at all.
"We're going to make a very good offer, we are going to be tabling it very soon, but there is a difficulty if you try to keep Northern Ireland in a customs union because one of the basic things about being a country is you have a single customs perimeter and a single customs union."
Formal proposals to resolve the dispute over the future of trade across the Irish border are expected to be tabled this week.
Responding to the RTE report on Monday night, Mr Coveney tweeted: "Non-Paper = Non-Starter. Time the EU had a serious proposal from the UK Govt if a #Brexit deal is to be achievable in October. NI and IRE deserves better!"
An Irish Government spokesman said a credible alternative to the backstop had yet to be proposed by the UK.
Shadow Brexit secretary Sir Keir Starmer said: "If Boris Johnson had spent any time listening to businesses and communities in Northern Ireland, he would know that these proposals are utterly unworkable."
He added: "If accurate, these proposals represent yet another failure of the Government's negotiating strategy.
"The Prime Minister should admit he has no credible plan for Brexit and that the only way to resolve this issue is to go back to the people with a public vote."
Meanwhile, The Times reported that Mr Johnson's plan to get around the Benn Act - the law aimed at preventing a no-deal Brexit without MPs' approval - would be to ask EU leaders to rule out any extension to the October 31 deadline.
Asked whether he could request a fellow Eurosceptic leader like Hungary’s Viktor Orban to veto an extension request, Mr Johnson told the BBC: “It’s a matter of common observation that the UK wants to come out. I don’t think any purpose is served in corralling the UK inside the EU”.
He added: “I think it would be a mistake to keep the UK bound in beyond the time people want to come out.”