Ms Merkel said if the UK could find a workable alternative to the backstop she would listen, while Mr Johnson welcomed the "blistering timetable of 30 days" and agreed the "onus is on us to produce those solutions".
But the PM was told the EU would be prepared for a no-deal Brexit as he was welcomed to Berlin by Ms Merkel.
On his first overseas visit, the German Chancellor told him she wanted a Brexit outcome which allowed the UK and EU to continue to have "very close relations".
Mr Johnson said a deal was possible but only if the EU dropped the backstop - the contingency plan aimed at preventing a hard border between the UK and Ireland.
It came after a formal welcome ceremony for Mr Johnson in the German capital, where he arrived to chants of "stop Brexit".
Mrs Merkel said at a press conference: "Britain is leaving the European Union. We have said repeatedly from the German perspective that we very much regret this but it is fact.
"So what we now need to do is to shape Britain leaving the European Union in such a way that we continue to have very close relations between the UK and the EU."
She added: "We are also prepared for a no-deal so, should this happen, this will or can happen, we are prepared for it.
"Obviously we also think of the many citizens, British citizens living currently in other countries of the European Union who also have to deal with the situation should Britain, from one day to the next, no longer be a member of the European Union."
Mr Johnson, whose stance on the backstop has been met with a wall of resistance in Brussels, insisted he wanted a Brexit deal.
But he added: "Clearly we cannot accept the current Withdrawal Agreement, arrangements that either divide the UK or lock us into the regulatory and trading arrangements of the EU, the legal order of the EU, without the UK having any say on those matters.
"So we do need that backstop removed. But if we can do that then I am absolutely certain that we can move forward together."
Mr Johnson said the "onus is on us" to produce solutions for the Irish border and he welcomed a "blistering timetable of 30 days" to come up with the answers.
He added: "I think what we need to do is remove it whole and entire - the backstop - and then work, as Chancellor Merkel says, on the alternative arrangements.
"There are abundant solutions which are proffered, which have already been discussed. I don't think, to be fair, they have so far been very actively proposed over the last three years by the British Government.
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"You (Mrs Merkel) rightly say the onus is on us to produce those solutions, those ideas, to show how we can address the issue of the Northern Irish border and that is what we want to do.
"I must say I am very glad listening to you tonight Angela to hear that at least the conversations that matter can now properly begin. You have set a very blistering timetable of 30 days - if I understood you correctly, I am more than happy with that."
Mr Johnson is meeting Ms Merkel amid warnings that she and other EU leaders will not make 11th-hour concessions over the Prime Minister's requests.
After talks with the German Chancellor, Mr Johnson will travel to Paris for talks with French President Emmanuel Macron.