Ahead of the first detailed negotiating sessions, Barnier and his opposite number Brexit secretary David Davis said that the menu for the talks would include Citizens' rights, the U.K.'s financial obligations to the European Union - known as the Brexit bill - separation issues, and Northern Ireland. "Now it's time to get down to work and to make this a successful negotiation", he added.
Also to be determined is the cost of Britain's exit from the bloc.
"We made a good start last month, and this week we'll be getting into the real substance", Davis said on July 16.
Barnier, who has repeatedly called on weakened British Prime Minister Theresa May to quickly set out her divorce strategy, said they needed to "examine and compare our respective positions in order to make good progress".
One of them hit back by claiming he was a member of "the establishment" which seeks to keep Britain in the European Union, claiming the Treasury sought to "frustrate" Brexit. Over the weekend he was subjected to briefings against him over a range of remarks he had allegedly made at a Cabinet meeting last week. "We'll now delve into the heart of the matter", Barnier told reporters, declining questions until the end of this round of talks on Thursday.
Foreign Secretary Boris Johnson then fanned the flames when he said in the Commons that Brussels could "go whistle" if it expected the United Kingdom to pay a hefty "divorce bill" in respect of its outstanding financial obligations.
Fox also said it would be "foolish" to go into the Brexit negotiations without being prepared to walk away, saying Britain's negotiating partners needed to believe Britain would do so rather than accept a bad deal.
Brexit hardliners in the Cabinet and on the Tory backbenches are furious with Mr Hammond for championing a two-year transition deal to cushion the impact of leaving the EU.
But as the talks get under way, it is reported that Chancellor Philip Hammond is being accused of Brexit treachery and trying to frustrate the UK's withdrawal from the EU.
"The plenary meetings [of the Brexit talks] will show us whether there is a realistic basis for agreement or whether the British government can not move at all because of its own problems", said Elmar Brok, a German centre-right MEP who helps coordinate the European parliament's position on Brexit.