The BBC in its report quoted the source as saying on Monday that Johnson spoke to the German chancellor earlier about the proposals he put forward to the EU, but she made clear a deal based on them was "overwhelmingly unlikely".
Merkel said a deal would never be possible unless Northern Ireland stayed in a customs union, the source added.
The source said the call between Johnson and Merkel was a "clarifying moment", adding: "Talks in Brussels are close to breaking down, despite the fact that the UK has moved a long way."
The latest development came after leaks from the European Commission showed major concerns from the European Union (EU) about the UK's Brexit plan.
Johnson sent new proposals to Brussels last week, with the Prime Minister urging the EU to enter detailed talks to "thrash out" any objections.
The main focus was to replace the so-called backstop - the policy negotiated by former Prime Minister Theresa May and the EU to prevent a hard border returning to the island of Ireland - which has long been a sticking point with Parliament for agreeing a deal.
Johnson has insisted the UK will leave the EU on the Brexit deadline of October 31, with or without a deal.
That is despite legislation passed by MPs last month known, as the Benn Act, which requires Johnson to write to the EU requesting a further delay if no deal is signed off by Parliament by October 19 - unless MPs agree to a no-deal Brexit.
( With inputs from IANS )