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”.
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”
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. Mr Barnier is expected to release a written statement on Mrs May’s speech later.