The UK and the EU will remain the "best of friends" but they will "not be as close as before" after Brexit, the new European Commission president has said.
Speaking ahead of talks with the PM, Ursula von der Leyen warned it would be "impossible" to reach a comprehensive trade deal by the end of 2020.
She said if the deadline was not extended it was not a case of "all or nothing", but of priorities.
Boris Johnson has insisted a deal is possible by December 2020.
After their meeting in No 10, a Downing Street spokesman said talks had been "positive", but the PM had been "clear" the process of negotiation would not be extended.
After its 31 January exit, the UK will enter into an 11-month transition period in which it will largely follow EU rules but will not have any representation in the bloc's institutions. This period will come to an end on 31 December.
Only when the UK leaves the EU can the two sides begin talks on their future economic relationship.
Mr Johnson told Mrs von der Leyen he "wanted a positive new UK and EU partnership, based on friendly co-operation, our shared history, interests and values", as well as a "broad free trade agreement covering goods and services, and cooperation in other areas".
He also said the UK was ready to start negotiations "as soon as possible" after 31 January.
Speaking at the London School of Economics earlier in the day, Mrs von der Leyen said the EU was "ready to negotiate a truly ambitions partnership with UK" but she warned of "tough" talks ahead.
"We will go as far as we can, but the truth is that our partnership cannot and will not be the same as before and it cannot and will not be as close as before because with every choice comes a consequences with every decision comes a trade off."
Mrs von der Leyen, a former German defence minister, took over from Jean-Claude Juncker at the start of December. She was a student at the LSE in the 1970s.
She also attended the same school as Mr Johnson in Belgium - something the prime minister highlighted as they posed for photos in Number 10.
Mrs von der Leyen said she hoped the new trading relationship would be based on "zero tariffs, zero quotas, zero dumping".
But she said: "Without the free movement of people you cannot have the free movement of capital and services.
"The more divergence there is the more distant the partnership will be."
Mrs von der Leyen also warned that without an extension of the transition period beyond 2020 "you cannot expect to agree every single aspect of our new partnership".
She called the deadline "very tight".
Opposition MPs have warned that trade deals typically take years to conclude and the UK risks defaulting to World Trade Organisation rules at the start of 2021, potentially leading to damaging tariffs for some industries.
But Brexit Secretary Stephen Barclay told BBC Breakfast the UK and EU had agreed in the political declaration to do a trade deal by the end of this year and he was "confident" they would do that.
'Expect red-line drawing with smiles'
The meeting between Boris Johnson and new European Commission President Ursula von der Leyen is significant in that it's their first face-to-face in their new roles - but today does not mark the start of post-Brexit trade talks.
EU law dictates that trade talks can't start until the UK legally leaves the bloc. Then EU countries must agree a mandate for the EU Commission to negotiate a comprehensive trade agreement on their behalf.
This mandate then has to be formally signed off at minister level by representatives of all EU countries.
All this means, the EU says, is trade talks will start at the beginning of March.
When UK ministers complain that's too long to wait, the EU response is that the UK always pushed for bigger role for national governments in EU decision-making to make it more democratic.
Expect red-line drawing with smiles today between the prime minister and Mrs von der Leyen - presented as "friends telling each other truths".
The EU position is that the prime minister's timetable to get an "ambitious, comprehensive" trade deal agreed and ratified by December is unrealistic.
However, the prime minister will counter this with "truths" of his own, including that negotiations have to be done by December because he won't extend the transition period.
Legislation implementing the terms of Mr Johnson's Brexit deal continues to move through the Commons, with the government easily winning all three votes on the Withdrawal Agreement Bill on Tuesday.
The bill will enshrine in law the terms of the transition period, first negotiated by Mr Johnson's predecessor Theresa May, as well as agreements on citizens' rights, customs arrangements in Northern Ireland and the UK's financial settlement.
Attempts by Northern Ireland parties to amend the bill to ensure "unfettered access" for businesses there to the rest of the UK market failed to pass on Wednesday afternoon.
MPs also rejected an attempt by Labour to reinstate child refugee protection rights in the Brexit bill.