But it is likely to accede to EU demands that the date should be when Britain leaves the EU in 2019, the FT said.
British Prime Minister Theresa May has called on the EU to issue a sweeping, reciprocal guarantee of rights for British expatriates living in the EU after Brexit. But EU leaders insist these must be negotiated in detail for them to have any legal value as reassurance for those people affected.
The FT, citing anonymous officials, said Davis would offer to guarantee the rights that EU citizens currently have in Britain, such as the freedom to move and work in the country, and aim to treat them "as fairly as they have been to this point".
The newspaper said one area of concern for the EU was whether those EU nationals living in Britain would be able to access the European Court of Justice, a "red line" for London.
Nine people killed in suicide attack on Mogadishu hotel
A spokeswoman for the 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."
Britain's preparations for the complex Brexit talks have been thrown into turmoil after Prime Minister Theresa May failed to secure a parliamentary majority in a national election last week.
The government is still in talks with a small Northern Irish party to secure the support of its 10 members of parliament to pass legislation.
More about: #Brexit