In a major speech in Florence on Friday, she will say history will judge Brexit "not for the differences we faced, but for the vision we showed".
The BBC understands she will propose a two-year transitional deal, after March 2019, ahead of a permanent trade deal.
It could include payments worth 20bn euros (£18bn) over the two years.
May 'to offer 20bn euros' transitional deal
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A guide to the Brexit negotiations
According to pre-released excerpts from her speech, the prime minister will say that a successful final agreement is in the interests of both the UK and the remaining EU countries.
"If we can do that, 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; not for the challenges we endured but for the creativity we used to overcome them; not for a relationship that ended but a new partnership that began."
She is expected to say, if the UK and EU can be "imaginative and creative" about establishing a new relationship, both sides can be "optimistic about the future".
Mrs May will argue that it is "in all of our interests for our negotiations to succeed… so 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".
Analysis
Do expect Theresa May's first admission that the UK will ask the EU for as long as two years to make our complete exit.
Two years during which we might pay billions to keep our existing ties with the single market. Two years, after 2019, that could give business breathing space.
But this could frustrate some voters who chose to leave, who may see departure now only in the distance.
Don't expect chapter and verse on the future relationship. There is not yet a clear agreement in cabinet on its shape and style, even if the prime minister herself had a fully fleshed-out vision.
But there is hope in government circles that the offer could unblock the Brexit negotiations.
There are also expected to be additional guarantees for EU citizens that could help, before the two sides are due to sit down again for official talks next week.
Read more from Laura
The BBC understands the prime minister will make what has been described as an "open and generous" offer, potentially worth 20bn euros over the two years - which could plug a black hole in the current EU budget, which runs to 2020 - created by the UK's departure.
Mrs May is not expected to say exactly how much the cash offer will be worth, or suggest that it is only being given in return for continued access to the single market in her much-anticipated speech, as these are subject to the negotiations in Brussels.
Additional long-term liabilities, like EU pensions and debts, will also have to be dealt with in the talks to come, so the eventual Brexit bill is likely to be far higher than that 20bn euros.
BBC political editor Laura Kuenssberg said the prime minister was also expected to go further than she had done previously on the issue of EU citizens' rights, which she is anticipated to say will be written into the withdrawal treaty. Whether that will be enough for the EU, which wants the European Court of Justice to be the ultimate guarantor of citizens' rights, remains to be seen.
The prime minister is also expected to repeat her assertion that the UK will seek its own bespoke trade deal after Brexit with the rest of the EU.
Stephan Mayer, a member of the German Bundestag for CSU party, told BBC Radio 4's Today programme that the EU was expecting more detail in Mrs May's speech about the UK's position - in particular about the proposed transition period and the status of EU citizens in the UK.
He added he believed the UK would not be able to have access to the single market without freedom of movement.
"I'm personally open for this proposal of a transition period but certainly you can't have one thing without the other," he said.
'Just for starters'
Conservative MP Kwasi Kwarteng, who campaigned for Brexit, said on BBC One's Question Time that he understood the "logic" behind a two-year transition period, when the current EU budget ran until 2020.
He acknowledged that to those who supported Brexit it was "frustrating" to have to continue paying into the EU for two years after leaving. But he said: "I think as a matter of goodwill, there is a sense in saying we will pay to the end of that budget process... and then not a penny more."
But Lib Dem Leader Sir Vince Cable, who was also on the panel, said: "£20bn is just for starters. It's going to be a lot more than that - it's an opening offer."
Labour MP Heidi Alexander told BBC Breakfast that a transitional deal was "absolutely essential" to provide certainty for businesses, while Tory MP Peter Bone told the programme that "it makes no sense" to give the EU "billions of pounds more".
The EU's chief Brexit negotiator Michel Barnier said on Thursday that striking a deal with the UK was "in our common interest" but he was awaiting "clear commitments" from the UK on the issue of guaranteeing EU citizens' rights, on the financial settlement and on the Northern Ireland border.
Without those issues being resolved in a withdrawal agreement, there would be no transition deal, he said.
The fourth round of Brexit negotiations begin on 25 September, with the UK due to leave the EU in March 2019.
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