The UK should make a unilateral decision to safeguard the rights of EU nationals living in the UK, according to a cross-party group of MPs.
They say EU citizens who contribute to the UK economy should not be forced to live "under a cloud of uncertainty".
Members of the Exiting the EU committee also want healthcare and pension rights for UK expats living overseas.
The government says it wants to guarantee the rights of EU citizens and British nationals "as early as we can".
Last week peers defied ministers by 358 votes to 256 to guarantee the rights of EU nationals living in the UK after Brexit - a move the government will seek to overturn when its Brexit Bill returns to the House of Commons.
Theresa May says she plans to trigger Article 50 - which kick-starts the formal process for the UK leaving the EU - by the end of this month.
But Hilary Benn, the Exiting the EU committee's Labour chairman, said it was wrong to use EU and British citizens as "bargaining chips".
"EU citizens who have come to live and work here have contributed enormously to the economic and cultural life of the UK," he said.
"They have worked hard, paid their taxes, integrated, raised families and put down roots.
"They did not have a vote in the referendum, but the result has left them living under a cloud of uncertainty.
"They are understandably concerned about their right to remain, and their future rights to access education and healthcare.
"Equally, Brits who live and work on the continent are worried about their right to work and access healthcare after Brexit...
"Although the government has said it wants EU citizens to be able to remain, this has not offered sufficient reassurance that the rights and status that they have enjoyed will be guaranteed - it should now do so."
In their report, 'The rights of UK and EU citizens', MPs urge all parties to the negotiations to make the resolution of the rights of all EU citizens in the UK, and UK citizens in the EU, their first priority.
The MPs argued "it would be unconscionable" for both sides "not to have clarity about their status for another two years".
"The government has made it clear it wants an early agreement to protect the rights of EU nationals in the UK, and of UK nationals living in other member states.
"The committee commend its commitment to this outcome - it is regrettable that this has not yet proved possible."
'Not fit for purpose'
Ministers have said it would be "unwise" to fully "guarantee" EU citizens' rights without a deal for Britons abroad.
Brexit Secretary David Davis said ministers had hoped to secure agreement on the issue at the European Council in December "but we couldn't get everybody to agree at that point".
The Exiting the EU committee report said the government needs to work out how to identify the estimated three million EU citizens who reply upon free movement as a basis for their right to remain.
It also argued that ministers will need to set a cut-off date for EU citizens arriving in the UK, adding: "Those who arrive before this date should retain the right to qualify for permanent residence after they have been here for five years.
"The government should decide and announce a date as soon as possible."
The MPs also called on the government to streamline the permanent residence system for EU nationals, describing it as "not fit for purpose", with applicants having to fill in an 85 page form "that is to complex and onerous to be used to clarify the status of up to three million people".
They warned that an abrupt reduction in the number of EU workers in the UK post Brexit "would cause disruption in a number of sectors".
"The government needs to take steps to train, or further incentivise training, to ensure that skilled workers are available to fill jobs in sectors currently featuring a large number of EU nationals," they said.
The committee also agreed there was a case for the UK to state during negotiations that it might be prepared to prioritise EU citizens in its future immigration policy - although former Conservative minister Peter Lilley dissented from this view.