Johnson, who supported the "leave" campaign, said that while there would be "lots of discussions about the nature of the deal, about money and so on, the most important thing is to raise our eyes to the future".
The British government wants the negotiations to include the future relationship with Europe and an all-important trade deal with the bloc.
But she now faces growing opposition at home to this, and her threats to walk away without a deal, in the wake of this month's general election in which she lost her centre-right Conservative party's parliamentary majority.
Talks will begin at 0900 GMT with a joint press conference by former French foreign minister and European commissioner Barnier and Davis at around 1630 GMT. "We will be fighting for all of these issues and for a final say when the talks are resolved".
Davis, who has previously said the discussion timetable would be "the row of the summer", is expected to initially accept the EU's approach, although a spokesman for his Department for Exiting the European Union said: "As the European Union has itself said, 'nothing is agreed, until everything is agreed'".
May's government said it was "confident it can achieve a bold and ambitious deal that will work in the interest of the whole U.K".
Sandro Gozi, Italy's European minister, had new doubts about Britain's negotiating position since the election result.
Brexit negotiations which could define the UK's political and economic future have begun, with David Davis calling for a "new deep and special partnership" between Britain and Brussels.
Without a parliamentary majority to support her controversial Brexit strategy, May's three main aims of leaving the single market, creating a new trade deal to replace customs union membership and withdrawing from the European Court of Justice are in danger.
Sterling also nudged higher to just under $1.28 and 87.42 pence per euro as formal negotiations got underway on Britain's exit from the European Union, which are expected to generate plenty of headlines for the currency in the weeks ahead.
Roth said that "Brexit is a very, very hard operation" and there's only a bit over a year to negotiate it.
"It seems freaky to press on before the government is secure in its ability to command a Commons majority and before the Queen's Speech, but the United Kingdom has to show willing - the dithering has to stop".
Mr Barnier's insistence on sticking to the EU's priorities for the negotiations comes after Mr Davis appeared to soften his stance on the schedule for the talks. He added: "Brexit won't make anything better, but it will make a lot of things more hard".
A diplomatic source said: "Doing a deal on Ireland and preserving the Good Friday agreement might be the way the Government quietly changes direction with a new agreement that will also be the template for the wider one with the European Union".