Mr Davis said he wanted to a secure "generous settlement" for EU migrants living in the UK and British citizens living in Europe.
But he warned setting a date now could mean a rush of people coming to Britain before any deadline - and any measures had to be within EU law.
Mr Davis told Sky's Murnaghan programme: "If we make a very generous settlement as I'd like to do, then people are going to say, 'Oh but then that'll attract lots more people in because they want to beat the deadline'.
"And so what I've said is, let's deal with that issue when we come to it.
"One way of dealing with it could be saying 'OK, only people who arrived before a certain date get this protection' - there are other ways too.
"But we've got to do it within the law as it stands because at that point we'll still be within the European Union."
He has been appointed by new prime minister Theresa May to oversee Britain's negotiations to exit the bloc following the referendum result last month.
Mr Davis reasserted his belief the EU would grant Britain access to the single market as well as a suspension of free movement rules, something which European leaders have so far ruled out.
And he has insisted Scotland cannot have a veto over any deal to leave the EU despite Mrs May suggesting all of the UK should agree a unified approach.
Mr Davis dismissed suggestions made by Scottish First Minister Nicola Sturgeon that Scotland could stay in the EU while the rest of the UK leaves.
His comments come after Australia called for a free trade deal with Britain as soon as possible in a Brexit boost for the Government.
Sky News.