The European Commission president has said Britain will need to have full movement of people post-Brexit - if it wants full movement of goods.
Ursula von der Leyen's comments come ahead of her visit to Ireland on Wednesday.
Addressing the European Parliament on Tuesday, she said: "It's the UK's choice to decide on how close or how distant they want to be from the European Union, but that also means from the single market.
"You cannot have no free movement for people, then expect to have free movement for goods, capital or services.
"It's either all four have a free movement or none of this is possible.
"You cannot expect to have a level playing field if there's a huge divergent in taxation, or social standards or environmental standards.
"So it is the choice of the UK how far they want to align or diverge - but this is decisive for how good the access to the single market will be or not.
"In short, it's the old proverb: you cannot have the cake and eat it at the same time".
She will be in Dublin on Wednesday to meet with An Taoiseach Leo Varadkar on her first official visit to Ireland.
The next stage of the Brexit negotiations is set to be high on the agenda.
The two will hold a joint press statement, before Mr Varadkar hosts a dinner for Ms von der Leyen.
She accepted the invitation which was extended by the Irish Government.
Ms von der Leyen recently visited London, where she warned that not every aspect of the EU and UK's future relationship will be able to be agreed on time if the Brexit transition period is not extended beyond 2020.
"The truth is that our partnership cannot and will not be the same as before.
"It cannot and will not be as close as before, because with every choice comes a consequence... with every decision comes a trade-off."
British MPs in the House of Commons passed the Brexit Withdrawal Agreement by a significant majority last week.