It would take the British province of Northern Ireland out of the EU's customs union but keep it largely aligned with the bloc's "single market" standards and regulations.
EU officials said this would not remove the need for customs checks -- a deal-breaker because it jeopardises the 1998 Good Friday Agreement that ended three decades of sectarian violence in Northern Ireland.
That agreement effectively created an invisible border between north and south, satisfying republicans who want a united Ireland and unionists who want to keep the status quo.
Keeping the border open and free-flowing has long been the main sticking point in the talks.
Downing Street officials say Brussels is making a big mistake because failure in the coming days to reach a deal would result in Britain's position only hardening down the line.
One source in Johnson's office told The Spectator magazine the government will try to "do all sorts of things" to prevent another Brexit delay should negotiations really collapse.