PM Theresa May has said there should be a transition period of "about" two years after Brexit, during which trade should continue on current terms.
EU migrants will still be able to live and work in the UK but they will have to register with the authorities, under her proposals.
And the UK will pay into the EU budget so member states are not left out of pocket.
She hopes this offer, made in a speech in Italy, will unblock Brexit talks.
The EU's chief Brexit negotiator Michel Barnier described the speech as "constructive" and said the prime minister had shown "a willingness to move forward".
The prime minister also proposed a "bold new security agreement" and said the UK would be the EU's "strongest partner and friend".
On trade, she said the two sides could do "so much better" than adopt existing models.
There was "no need to impose tariffs where there are none now", the prime minister said.
She did not mention how much the UK would be prepared to continue to pay into the EU for two years after it leaves in March 2019, but it has been estimated as being at least 20bn euros (about £18bn).
She also suggested that the UK and EU would continue working together on projects promoting long-term economic development and the UK would want to "make an ongoing contribution to cover our fair share of the costs involved".
When the two-year transition period is up, the UK and EU could move towards a new "deep and special partnership," she said in her speech.
To EU citizens in the UK she offered reassurance that "we want you to stay, we value you" and acknowledged differences with the EU over which courts should guarantee their rights after Brexit.
She said she wanted UK courts to take account of rulings by the European Court of Justice and hoped "on this basis, our teams can reach firm agreement quickly".