PM Theresa May has said there should be a two-year transition period 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 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.
She 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 after Brexit, but it has been estimated as being at least 20bn euros (about £18bn).
In her speech, Mrs May said the UK would “honour commitments” made while it had been a member to avoid creating “uncertainty for the remaining member states”.
She also suggested that in the long-term the UK and EU would continue working together on projects promoting 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 towards a new “deep and special partnership,” she said in her speech.
By March 2019, neither the UK or EU would be ready to “smoothly” implement new arrangements needed: “So during the implementation period access to one another’s markets should continue on current terms and Britain also should continue to take part in existing security measures.”
Such a period should be “time limited”, she said, as neither the EU nor the British people would want the UK to remain in the EU longer than necessary.
But she hoped to build a “comprehensive and ambitious” new economic partnership with the EU in the long-term.
This should not be based on existing agreements with Canada or European Economic Area membership, she said, but a “creative solution” should be found to reflect the existing relationship between the UK and EU.
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”.
Mrs May opened her speech by saying Brexit was a “critical time in the evolution of the relationship between the United Kingdom and European Union”.
She said if “we open our minds to new thinking and new possibilities we can forge a brighter and better future for all our peoples”.
Some voters were worried about the prospect of Brexit – but others found it an “exciting time,” she said.
“I look ahead with optimism, believing that if we use this moment to change not just our relationship with Europe but also the way we do things at home – this will be a defining moment in the history of our nation.”