What is the problem?
Both the European Union and the UK have said they want to keep a soft border between Northern Ireland and Ireland because of the closely interlinked trade between the two jurisdictions and the provisions of the Good Friday Agreement.
Why has Brexit caused this difficulty?
After leaving the EU, the UK will be able to diverge from Brussels’ regulations and the tariffs levied on imports to the bloc.
This will mean that Northern Ireland and Ireland could have substantially different systems in operation and checks will be required on goods crossing the border.
A contingency plan included in the Withdrawal Agreement – the divorce deal negotiated with the EU by Theresa May’s government – in order to keep a soft border unless and until other options are found.
Mr Johnson has called for it to be scrapped – a “backstop-ectomy” as he put it – because he claims the measure would, potentially indefinitely, leave the UK trapped in a customs union with the EU and result in Northern Ireland being bound by single market rules over which it has no influence.
So what’s Mr Johnson’s plan?
He has yet to set out in detail how he proposes to get around the problem, but used his meeting with Mrs Merkel to highlight the report drawn up by the Alternative Arrangements Commission, a panel established by the Prosperity UK think tank.
Do his staff believe he can achieve a deal?
The minister for school standards told ITV News he welcomes Angela Merkel's "acceptance that there are alternatives to the backstop".
Nick Gibb says that means Britain is now able to "negotiate effectively with the European Union" and believes the German chancellor's statement "increases the likelihood that we will leave with a deal".
Despite claiming that a deal can be achieved and a backstop alternative can be found, Mr Gibb did not reveal any specific plan to ensure either of those outcomes.
He said: "We will negotiate rigorously and sincerely with the European Union to try and secure a deal and an alternative to the Irish backstop."
What could that these alternative arrangements involve?
That commission, led by senior Tories Greg Hands and Nicky Morgan, produced a 268-page report in July which recommended a series of administrative and technological measures.
Under their plans special “enhanced economic zones” could straddle the border between Londonderry and Donegal and possibly also Newry and Dundalk, with tax breaks and a free trade zone to avoid duties.
If the regimes on plant and animal regulations diverge, mobile units could be used to carry out sanitary and phytosanitary (SPS) checks away from the border.
Trusted trader regimes would allow firms to cut down on paperwork and potentially avoid routine customs checks at the border.
Would the EU be prepared to accept alternatives to the backstop?
As the backstop is only intended as a fallback option, Brussels is willing to explore alternatives to prevent it being implemented.
But the EU insists that the legally binding insurance policy is required should those alternatives fail to meet the requirements or be ready in time for the end of any transition period contained in a final Brexit deal.
So is there hope for a breakthrough?
Unless Mr Johnson backs down on his demand for the backstop to be scrapped, or Brussels budges on its insistence that a contingency plan should be in the legally-binding Withdrawal Agreement, the two sides appear to be at an impasse.