The development comes a day before Johnson is slated to speak to French President Emmanuel Macron, German Chancellor Angela Merkel and European Council President Donald Tusk on Monday "to see if there's an agreement on a 'landing zone' for Northern Ireland and customs', the Daily Mail quoted a government source as saying to the Sunday Times.
Both sides were now in intensive negotiations on a new "best of both worlds" blueprint thought to involve Northern Ireland technically staying in the UK's customs union - but Britain collecting tariffs on the EU's behalf.
It should become clear by Monday whether an agreement was possible and it would be finalised at a crucial summit in Brussels on October 17 and 18.
Writing in The Sunday Telegraph, Commons Leader Jacob Rees-Mogg said: "In the final stages of the Brexit negotiation, compromise will inevitably be needed, something even the staunchest Leavers recognise albeit unwillingly - but as a Leaver Boris can be trusted.
"He wants to take back control and has dedicated his political career to this noble cause..."
British government officials have so far remained tight-lipped over the nature of the new Brexit proposal, The Daily Mail said.
According to speculations, it would create a customs border in the Irish Sea with goods travelling from the rest of the UK to Northern Ireland being subject to tariffs which Britain would collect on behalf of the EU.
Businesses would then be able to claim a rebate once they had shown the goods were for consumption in the UK market.
However it would mean that Northern Ireland would be able to benefit from any post-Brexit trade deals the UK struck with other countries around the world.
On Friday, Tusk said that he received "promising signals" from Irish Prime Minister Leo Varadkar that a Brexit deal was still possible.
Tusk's remark came after Johnson and Varadkar held "detailed and constructive" talks for more than two hours on Thursday, with the latter saying the "very positive" meeting meant negotiations could resume in Brussels, the BBC reported.
The UK is due to leave the EU at 11 p.m. (local time) on October 31.
With three weeks to go until the Brexit deadline, Johnson continues to insist the UK will leave the EU with or without a deal at the end of the month.
That is despite the so-called Benn Act - passed by MPs last month - demanding he request a delay to the Article 50 deadline from the EU until January 2020 if a deal has not been agreed before October 19.
Should any new deal be reached between the EU and UK, it will still have to be approved by MPs in Parliament.
( With inputs from IANS )