The Brexit Secretary is hoping to win round Michel Barnier over Britain's plan to guarantee the future rights of European nationals living in the United Kingdom amid deep divisions back home.
"It's time to get down to work and make this a successful negotiation", veteran anti-EU campaigner David Davis said as he was welcomed by the bloc's chief negotiator Michel Barnier before their teams began four days of talks.
May faces the daunting task of navigating Brexit after being stripped of her majority in the House of Commons in June's election, while holding her Conservative government together amid squabbling ministers and talk of a potential putsch. This was "more a technical argument" and still had to be negotiated with the EU, Fox said.
Mr Hammond lifted the lid on cabinet feuding by claiming damaging stories about him had come from fellow ministers out to get him because he is pushing for a softer Brexit.
European officials have said the British proposal to give EU citizens "settled status" does not go far enough.
The EU reportedly wants Britain to pay from $70 billion to more than $110 billion to cover ongoing EU budget commitments, an amount the British have rejected.
Earlier on Sunday, Chancellor Philip Hammond had said senior British government ministers were becoming convinced of the need for transitional arrangements to reduce disruption as Britain leaves the EU.
"We made a good start last month, and this week we'll be getting into the real substance", Davis was quoted by Reuters as saying ahead of the meeting.
Davis and Barnier spent only a half-hour in negotiations Monday, and photographs from the talks featured bulging files of papers on Barnier's side of the table and little evidence on Davis'.
But May's minority government remains fragile one month after the snap June 8 election in which her Conservative Party lost its majority, leaving the European Union wondering whether she can actually formulate a coherent Brexit policy. "This government is facing a ticking clock over the Brexit negotiations", Hammond said on Sunday.
Johnson, who was in Brussels on Monday separately to meet his 27 European Union colleagues, said he hoped the bloc would accept the "very fair and serious offer" Britain had made on the rights of European Union citizens.
Negotiators will focus on key issues concerning Britain's withdrawal, including citizens' rights, Britain's exit bill and the border in Northern Ireland.