Video report by ITV News Europe Editor James Mates
The president of the European Commission has said the EU is "ambitious" that a new, "unprecedented" trade deal can be reached with the UK by the end of 2020.
But , who was recently elected to replace Jean-Claude Juncker at the helm of the Commission, warned it would be "impossible" to agree a full comprehensive deal within the 11 month transition period which begins when the UK leaves the EU on January 31.
Following the meeting a Downing Street spokesman described the discussions as "positive" and said the PM planned to start negotiating a "Canada-style FTA as soon as possible after January 31".
"The Prime Minister has been clear that the implementation period will end at the end of this year and we will work with the European Union to negotiate an FTA (free trade agreement)," added the Number 10 spokesman.
But Ms von der Leyen conceded that "without an extension of the transition period beyond 2020, you cannot expect to agree on every single aspect of our new partnership".
"We are prepared to design a new partnership with zero tariffs, zero quotas, zero dumping - a partnership that goes well beyond trade and is unprecedented in scope," she added.
While in Number 10, Ms von der Leyen and the prime minister laughed about having went to the same school in Belgium and reminisced about a teacher they shared.
The PM said: "Ursula and I were at school together - we were at the same school."
Mr Johnson joked about introducing "a very rough game" to the school called "British Bulldog" - "it's a very good game," he added.
Ms von der Leyen said January 31 - Britain's exit day - would be a "tough and emotional day".
But she added: "When the sun rises again on February 1, the United Kingdom and the European Union will still be the best of friends and partners."
She ensured the audience that ties between the UK and the European Union will continue to be "unbreakable", but she said there would be "consequences" to Brexit.
"Our partnership cannot and will not be the same as before," she said.
"It will not be as close as before because with every choice comes a consequence. With every decision, comes a trade-off.
"Without the free movement of people, you cannot have the free movement of capital, goods and services.
"Without a level playing field on environment, labour and state aid, you cannot have the highest quality access to the world's largest single market.
"The more divergence there is, the more distant the partnership will be."
He explained that today's Downing Street meeting will focus on how the two sides can "ensure we do have that trade deal in place" but said fine details won't be discussed.