The differences between the two leaders mainly revolve around the safeguard or "backstop" for Ireland, which would obligate the United Kingdom to remain a part of the EU Customs Union although it would not define the future relations with the European Union after Britain's exit from the bloc.
"The backstop has always been a fall-back option until this issue is solved and one knows how one wants to do that," Merkel said on Wednesday before she discussed the matter with Johnson in Berlin, Efe news reported.
"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?" she added.
The Irish government says that maintaining the backstop is a crucial element of any British exit from the EU.
Johnson, meanwhile, told Merkel that he wanted a speedily negotiated Brexit deal that does not include the backstop calling it "undemocratic" so that the UK can leave the EU with some kind of deal prior to the October 31 deadline.
"We do need that backstop removed," said Johnson. "If we can do that, then I'm absolutely certain we can move forward together."
Despite their conflicting stances, Johnson was optimistic that a mutually agreeable solution could be found, noting that often negotiations with Europe have seemed to have no way out but ultimately an accord has been reached.
He likened the situation to a horse race, saying that everything often comes down to the final leg of the race when a solution is actually achieved.
Merkel said that she wanted Britain to set forth its proposal on the backstop issue.
"We will first listen to Britain's proposals. Our goal is to preserve the integrity of the single market," she said, going on to express optimism by adding, "And just as we managed to discuss and solve issues with imagination within the European Union, I believe that you can also find ways here and this will be the task."
"And to that extent, I would say from the German side, and this is what we'll be talking about today, that we would welcome a negotiated exit from the EU, of course, but we have repeatedly said that we are also prepared if such a negotiated exit should not come to fruition," the German leader added.
Johnson assured Merkel and the public that London will respect the rights of European citizens living in the UK.
He also tried to set a relaxed tone in the joint news conference, recounting the many spheres in which Germany and the UK have worked together successfully from security to biodiversity emphasising that all that will be part of the agenda of his visit.
"We have to maintain the integrity of the single market," Merkel said.
( With inputs from IANS )