- Video report by ITV News Europe Editor James Mates
Theresa May will tell European counterparts their shared political legacies depend on agreeing a good Brexit deal as she seeks to break the deadlock in negotiations over the UK's withdrawal from the EU.
On Friday, the prime minister will set out her plans for a transitional period from the formal date of Brexit in March 2019, expected to last two years, before moving to a permanent trade deal.
ITV News understands Mrs May is understood to be considering an offer of paying around £20 billion over the transition period in order to secure favourable trade access, which would also help address Brussels' concerns about the potential hole in its current budget caused by the UK's exit.
Her landmark speech comes as an opinion poll suggested a majority of Britons now back staying in the European Union - with 52% in favour of remaining part of the bloc.
In the Italian city of Florence, Mrs May will remind European leaders they have a "profound sense of responsibility" to agree good terms.
Her direct pitch to the continent's leaders will be seen as a further attempt to try to bypass the formal talks led by Brussels' Brexit chief negotiator Michel Barnier, which have stalled amid concerns of a lack of progress on issues including the financial settlement, citizens' rights and the Northern Ireland border.
Mrs May will emphasise the historic nature of the negotiations as she calls for an "imaginative and creative" approach to securing a deal, in what may be seen as a fresh British complaint about the European Commission's perceived rigid approach to talks.
But Mrs May - who has repeatedly vowed that she would walk away with no deal rather than accept a "bad deal" from Brussels - will also stress that the UK has "considerable" fundamental economic strengths and an "indomitable spirit".
The Prime Minister will say that if a deal can be reached "then when this chapter of our European history is written, it will be remembered not for the differences we faced, but for the vision we showed" and "not for a relationship that ended but a new partnership that began".
She will acknowledge that Brexit is "inevitably a difficult process" but it is in "all of our interests for our negotiations to succeed".
"I believe we share a profound sense of responsibility to make this change work smoothly and sensibly, not just for people today but for the next generation who will inherit the world we leave them," she will say.
"The eyes of the world are on us but if we can be imaginative and creative about the way we establish this new relationship ... I believe we can be optimistic about the future we can build for the United Kingdom and for the European Union."
The Prime Minister's speech, which will build on the Brexit plans she set out in Lancaster House in January, will set out her vision for a "bold" economic and security partnership with a "time-limited" implementation period to avoid a cliff-edge change for businesses adjusting to the new arrangements.
But she is thought to have rejected a Swiss-style "European Economic Area minus" deal, which could have involved ongoing payments to Brussels and potentially accepting the free movement of people.
Foreign Secretary Boris Johnson, whose intervention on Brexit last week led to speculation he might be about to resign, was opposed to any continued payments for single market access after the transitional period expires.
After a marathon two-and-half hour session of the Cabinet on Thursday, Mr Johnson and Chancellor Philip Hammond - who have been at loggerheads all summer over Brexit - left No 10 together in an apparent show of unity.
In a message about the strength of the UK, the Prime Minister will use her speech to insist that "Britain's future is bright".
"Our fundamental strengths are considerable," she will say, including "a legal system respected around the world; a keen openness to foreign investment; and enthusiasm for innovation; an ease of doing business; some of the best universities and researchers you can find anywhere; an exceptional national talent for creativity; and an indomitable spirit".
On the eve of Mrs May's speech, Mr Barnier issued a fresh warning that Britain must "settle the accounts" if it wants a free trade deal when it leaves the bloc.
Mr Barnier said there was still "major uncertainty" over the UK's approach on key issues and a "question of trust" about the financial settlement.
Speaking in the Italian parliament, Mr Barnier - who has made clear he is not prepared to open negotiations on a free trade deal until the issue of a financial settlement is agreed - said he would be listening "attentively and constructively" to what she had to say.