Mrs May reportedly hopes that this financial offer will break a three-month deadlock in Brexit talks and allow negotiations to continue between David Davis and Michel Barnier.
EU officials are keen to pin Britain down over the exit bill, which is stalling talks from progressing.
According to the Financial Times, Olly Ribbons, the Prime Minister’s top EU adviser, has contacted his colleagues in European capitals to tell them that Mrs May’s Friday speech in Florence would include the financial offer.
The Prime Minister is due to fly to Florence this week to discuss her vision for a post-Brexit Britain and the future ties with Europe.
One EU official told the paper: “There is a contribution to make for that single market participation which is independent and on top of any debts from the past.”
British officials said it is likely that Mrs May has already told her cabinet of the details of her Florence speech.
UK officials have claimed Britain would ensure no member state would be required to pay more into the EU budget or receive less money from it until 2020.
In a 4,000 word article in the Telegraph, Boris Johnson told Mrs May to forget about paying excessive sums into the EU after 2019.
There is speculation that Mr Johnson could resign over Mrs May’s strategy on Brexit.
The Foreign Secretary believes the UK should have a loose relationship with the single market, similar to Canada’s free-trade deal.
Last Friday, the Brexit Secretary David Davis said that Britain’s EU divorce bill is “£60 billion and falling”.
Mujtaba Rahman, head of Europe for the Eurasia Group risk consultancy, said: “The message from the EU is clear: covering budget holes is only a start; for a deal in October May also has to acknowledge that the UK will pay to settle past debts.”