European Council President Donald Tusk said the UK could not pick and choose.
The French and German leaders also made clear that the freedom of movement of EU citizens was non-negotiable.
Immigration to the UK, particularly from poorer EU countries, was a key issue in the referendum campaign.
Some campaigners for Leave sent a clear message that the vote was about controlling immigration levels.
Outgoing UK Prime Minister David Cameron said that the issue of freedom of movement would be for the next PM and government to decide.