MPs will meet the EU's chief negotiator later to call for UK citizens' rights to be protected in a no-deal Brexit.
The cross-party delegation will meet Michel Barnier in Brussels, and be led by Tory MP Alberto Costa.
He said government could protect the 3.6 million EU nationals in the UK, but did not have the power to do the same for 1.3 million UK citizens in the EU.
Mr Costa said, as it stands, a no-deal Brexit would "terminate the rights of British citizens overnight".
The meeting comes after MPs voted on Thursday to prevent the next prime minister from suspending Parliament in order to push through a no-deal Brexit.
Four cabinet ministers, including Chancellor Philip Hammond, abstained, while 17 Tory MPs rebelled against the government.
Foreign Secretary Jeremy Hunt has discounted the possibility of suspending Parliament - known as prorogation - if he becomes prime minister, but his rival in the Conservative Party leadership race, Boris Johnson, has refused to do the same.
Health minister Stephen Hammond - who abstained in Thursday's vote - said he would not rule out voting to bring down his own government if its policy became pursuing a no-deal Brexit, telling BBC Radio 4's Today programme: "A lot of people were taught that you must put the interest of the country before yourself."
But leading Brexiteer and Tory MP Jacob Rees-Mogg told the same programme those trying to prevent it were "not saying what they believe", and their intention was to "snub the British voters" and stop Brexit altogether.
Mr Costa resigned from the government in March to put forward an amendment to Prime Minister Theresa May's Brexit bill, calling for her to write to the EU to demand protections for UK and EU expats' rights if there was a no-deal Brexit.
It ended up gaining government support, and was passed by MPs.
As a result, Brexit Secretary Stephen Barclay wrote to Mr Barnier, saying there was growing support for a ring-fenced citizens' rights agreement in the European Parliament, and in June, he visited Brussels to make the case for a separate agreement in case of a no-deal scenario.
But the EU has refused to "negotiate mini deals", insisting the best way to safeguard citizens' rights was to implement the withdrawal agreement negotiated between the bloc and Mrs May - which has been voted down by MPs three times.
Mr Costa told Today the government would be "abrogating its responsibilities" to UK citizens if it did not make bilateral agreements with the different EU countries.
"In the event of no deal, the United Kingdom Parliament can take measures to protect EU nationals in the UK, but we do not have powers to pass legislation extraterritorial, in other words within the EU, to protect our own citizens," he said.
"If Britain chooses to exit without an agreement in place, it would be terminating the rights of British citizens overnight."
He added: "I want to understand from Michel Barnier what his position is in carving out citizens' rights, why he has said, thus far, no to that."
On the table
The current deadline for the UK to leave the EU is 31 October, but if it fails to agree a deal to do so, the legal default is to leave with no deal on that date.
Both contenders to be the next prime minister said they want to keep to the date and renegotiate with the EU.
Mr Hunt and Mr Johnson have also said they would keep leaving without a deal on the table to strengthen negotiations, despite Parliament voting to rule the option out.
The EU has consistently said the withdrawal agreement is closed and cannot be changed.
The incoming president of the European Commission, Ursula von der Leyen, has said she hopes the UK remains in the EU, but it was up to British authorities to "sort its side of things on Brexit".
Asked about Mrs von der Leyen's position, German Chancellor Angela Merkel said on Monday she understood that "if the UK wishes for more time [e.g. another delay to Brexit], then they would have more time", but it was "up to the UK".
She praised her "cooperative relationship" with Theresa May, adding: "She has not had an easy time dealing with this difficult question. I always found her to be a reliable and collegial partner, and I thank her for that."
But she added: "We now have the position that there will be a new PM, and then we have to watch what that person decides."