Davis expects in return the EU to drop its demand for the European Court of Justice to supervise the rights of EU nationals after Brexit.
A spokesperson for the UK Brexit department said, "We have said consistently that resolving the status of EU citizens living in the UK and UK nationals living in other member states is our first priority for negotiations."
The EU Brexit negotiators, including Michel Barnier, have been pressing for citizen rights issues, along with the UK's divorce bill and the Irish border, to be dealt with immediately before discussions move to trade related matters and the UK's future relationship with the EU.
This is something Guy Verhofstadt, Parliament's lead Brexit negotiator, has consistently called for since the UK triggered article 50.
Speaking in plenary in Strasbourg this week, the Belgian MEP also said he believes that Britain is free to change its mind and stay in the EU, but that it would have to give up special perks including the hard-fought budget rebate.
Verhofstadt picked up on comments made by the French President Emmanuel Macron on Tuesday that the door to the EU would remain open to Britain during Brexit negotiations.
"I agree," he said, "but like Alice in Wonderland, not all the doors are the same. It will be a brand new door, with a new Europe, a Europe without rebates, without complexity, with real powers and with unity."
Meanwhile, in another twist to the looming Brexit talks, Scotland's First Minister Nicola Sturgeon has called on the UK government to "adopt a much more inclusive process in reaching UK objectives for the negotiations," which would include participation of the devolved governments in the negotiating team.
The demand comes in a letter to the UK Prime Minister Theresa May.
Sturgeon also asked the Prime Minister to maintain Britain's membership of the EU single market to "bring clarity, in place of the current confusion, and provide a coherent base for the UK's future relationship with the EU," arguing that May's proposal to leave the single market "failed to garner support" at the general election.
She added, "It is now clear that a new proposal is needed urgently to protect the economy and bring people together."