European nationals in the UK and British citizens living in the EU must have a guarantee they can remain in place after Brexit, campaigners say.
The Three Million group is demanding assurances people will not be used as "bargaining chips".
Meanwhile, the think tank British Future wants a "fair" cut-off date for any settlement and citizenship changes.
The government says it wants to let the 2.8 million EU nationals stay in the UK, but member states must reciprocate.
The Three Million group delivered a letter to Downing Street on Monday, alongside 10 groups representing the 1.2 million UK citizens living in the EU.
It wants EU citizens living in the UK to be given permanent residence before Article 50 of the Lisbon Treaty - getting Brexit negotiations with the EU under way - is triggered, which the government says will happen by the end of March.
"We are not bargaining chips," the letter reads. "We are people."
Speaking outside Downing Street, The Three Million's chairman Nicolas Hatton, a French former marketing worker who has lived in the UK for 21 years, warned Prime Minister Theresa May she would face a "bureaucratic nightmare" to register all the people unless she acted promptly.
The report by British Future, which describes itself as an "independent, non-partisan think tank", recommends the triggering of Article 50, as "a fair and legally watertight cut-off date", after which changes to the settlement and citizenship rights of other EU nationals "might apply".
It calls for the current EU permanent residence system to be converted into "indefinite leave to remain" status, which is already available to other migrants living in the UK.
But it also says the process should be streamlined and costs capped.
British Future's report says it is "morally wrong to use EEA (the EU plus Norway, Iceland and Liechtenstein) nationals in the UK as bargaining chips to secure the rights of UK nationals in Europe".
Labour MP Gisela Stuart, who was chairwoman of Vote Leave during the referendum campaign and chaired the British Future panel, which included a cross-party group of MPs, said: "Britain should make clear at the start of the Brexit negotiations that EU citizens already here before that date can stay.
"This would send a clear signal about the kind of country the UK will be after Brexit and the relationship we want with Europe."
Ms Stuart said EU nationals who have been in the UK for more than five years should have a fast-track process for acquiring UK citizenship.
She told BBC Radio 4's Today programme: "It's a question of who makes the first offer - then we can start these negotiations."
Ms Stuart added: "It must be right that people are free to plan their lives."
A government spokesman said: "The prime minster and other ministers have been absolutely clear that they want to protect the status of EU nationals already living here.
"The only circumstances in which that wouldn't be possible is if British citizens' rights in European member states were not protected in return."