The UK prime minister Theresa May has set out proposals for a two-year transition period after Britain leaves the EU in March 2019.
She wants existing EU market access arrangements to apply during that period and promised Britain would pay its "fair share" into the EU budget.
She said the UK will be the "strongest friend and partner" of the EU after Brexit.
The speech in Italy was aimed at breaking the deadlock in Brexit talks, due to resume on Monday.
The deal she proposed could include payments worth 20bn euros (about £18bn) over the two years.
The PM proposed a "bold new strategic agreement" on security co-operation and said she was optimistic about a "defining moment" in UK history.
On trade, she said the two sides could do "so much better" than adopt existing models and there was "no need to impose tariffs where there are none now".
In a wide-ranging speech, Theresa May also said:
- A "period of implementation" - potentially of two years - should be agreed "as early as possible"
- That the UK was prepared to "honour commitments we have made" - a reference to financial commitments
- And that it would continue to make "an ongoing contribution" to projects it considers greatly to the EU and UK's advantage, such as science and security projects
- She wanted a "bold new security relationship" with the EU which would be "unprecedented in its depth"
- That the UK did not want to "stand in the way" of closer EU integration, as outlined by Jean-Claude Juncker.
Negotiations are due to resume on Monday but so far the two sides have not reached agreement on the issues of the rights of EU and UK citizens, the financial settlement or the Irish border issue.
It is an offer, not a blinding revelation, but a limited flash of ankle to her continental counterparts.
Today was not about lavishing detail on the EU side who are eager to understand more about what it is the UK actually wants from them after we leave.
It was the notional writing of an undated check from us to them, that government insiders hope means next week, when the official talks get back under way, some progress can actually be made.
Theresa 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."
The 20bn euro offer is meant to ensure no EU countries are left out of pocket by Britain's departure.
But it is not part of the "divorce bill" covering the UK's outstanding debts and liabilities to the EU, which will still have to be agreed with EU negotiators, meaning the final bill for Brexit could be far higher.
The EU's chief Brexit negotiator Michel Barnier said on Thursday he was awaiting "clear commitments" from the UK on the citizens' rights issue, the divorce bill and the Northern Ireland border.
Without those issues being resolved in a withdrawal agreement, there would be no transition deal, he said. Barnier is expected to release a written statement on May's speech later..