Mr Davis travelled to Brussels ahead of the second round of formal talks.
He said his priority was to “lift the uncertainty” for EU citizens living in the UK and Britons living in the EU.
The EU says there must be substantial progress on this – and on a financial settlement and the issue of the Irish border – before trade talks can begin.
In a brief appearance alongside EU chief negotiator Michel Barnier as their teams began this week’s talks, Mr Davis said there had been a good start to the process and it was time to get to the “substance of the matter”.
Mr Barnier said the negotiators would “now delve into the heart of the matter”.