With 30 days to go until Britain is due to leave the European Union on October 31, Johnson is racing for an agreement to avoid yet another delay. But the bloc's leaders have complained they have yet to see a concrete alternative to the current divorce deal and time is running out, AFP wrote.
"We are going to make a very good offer. We will be tabling it formally very soon," Johnson told BBC television from Manchester, where his Conservative party's conference is underway.
He declined to give details but media reports said they could be proposed as early as Thursday.
Johnson is seeking to renegotiate the divorce terms struck by his predecessor Theresa May last year but rejected three times by the British Parliament.
He is focusing on the most controversial element, the so-called backstop plan to keep Britain subject to EU customs rules to allow goods to flow freely between British Northern Ireland and EU member Ireland after Brexit.
Irish broadcaster RTE reported late Monday that Britain has proposed erecting "customs clearance sites" along both sides of the border but located five to ten miles (eight-16 kilometers) away to keep the actual frontier open.
Citing a "non-paper" provisional plan put forward by London, it said goods moving from one site to another would be monitored in real time on mobile phones or tracking devices placed on trucks.
But Ireland says there cannot be a return to the border infrastructure of the past, warning it could upset the peace process in Northern Ireland.
Foreign minister Simon Coveney swiftly dismissed the leaked proposal as a "nonstarter", a position echoed by the Irish opposition party Sinn Fein.
Norbert Rottgen, chairman of the foreign affairs committee in Germany's Bundestag lower house of Parliament, said it was "not serious".
Johnson suggested the leak was a previous draft of the plan, insisting: "That's not what we're proposing at all."
However, he said it was "just the reality" there would have to be checks somewhere after Britain leaves the EU's customs union and single market.
Johnson took office in July promising to leave the EU on October 31 no matter what after May twice delayed Brexit in her efforts to get a deal.
His pledge is popular with Conservative members and many Brexit voters but MPs in the House of Commons fear a "no deal" exit would be disastrous.
They passed a law requiring him to ask the EU to delay again if he has not reached a divorce deal by a Brussels summit on October 17/18.
"We now have two weeks left to where there are no credible solutions on the table," Ireland's Europe Minister Helen McEntee told RTE.
"The ball is very much in the court of the UK, as it has been for months and months at this stage. We are trying to be as accommodating as possible."
The EU said it had yet to receive any formal proposals but one official said: "We are ready to examine workable proposals that achieve all objectives of the backstop."
Johnson insisted he was working "very hard" to get a deal and said progress had been made.
He said London had recently made "a big concession" in agreeing that Northern Ireland and Ireland would follow the same EU rules on agriculture, such as the movement of cattle.
If Johnson gets a deal, it must then pass the House of Commons, where he has no majority and faces significant hostility.
He has riled pro-European MPs with his accusations they are "surrendering" to Brussels and "betraying" the 2016 referendum vote for Brexit.
Hardline eurosceptic MPs have warned they would also reject any deal that does not deliver a clean break.
"We can't be allowed to become some kind of regulatory satellite of the EU – that is not the point of leaving," MP Steve Baker said.