"The backstop has always been a fall-back option until this issue is solved," she said.
"It was said we will probably find a solution in two years. But we could also find one in the next 30 days, why not?"
Mr Johnson replied: "You rightly say the onus is on us to produce those solutions, those ideas and that is what we want to do.
"You have set a very blistering timetable of 30 days - if I understood you correctly, I am more than happy with that," he added.
He added that alternatives to the backstop had not been "actively proposed" under his predecessor Theresa May - but he was pressed by Mrs Merkel to spell out what such alternatives might look like.