The civil society group behind the initiative, New Europeans, has called on the EU institutions to guarantee the rights of the 1.6 million Britons working in mainland Europe and the 3.4 million EU citizens living in the UK.
The campaigners have sent a letter to European Council President Donald Tusk calling for an end to citizens "being used as bargaining chips" in the Brexit negotiations.
Talks on Brexit will not start until after the UK election on 8 June but the rights issue has already been flagged up by Guy Verhofstadt, Parliament's Brexit negotiator, as a matter of urgency.
UK Socialist MEP Julie Ward, who backs the campaign, said, "UK citizens living in the EU are currently being used as pawns in Brexit negotiations, which is unacceptable. Individuals and families who have made their homes in other EU countries are being left in a cruel limbo while the negotiators spar back and forth.
"The EU is the world leader on human rights so it needs to put its head above the parapet and guarantee the unilateral recognition of the rights of UK citizens living in other EU countries."
Speaking at a news conference in Brussels earlier this week, former MP Roger Casale, CEO of New Europeans, called on the EU to safeguard the rights of UK citizens.
He said, "As committed Europeans, let us take a moment to reflect. We are building a 'Europe of the citizens', an EU that safeguards human rights. The EU is doing its best to protect the rights of 3.4 million non-British EU citizens in the UK and 1.6 million British EU citizens in other member states.
"British citizens resident in the EU are not UK expats, they are EU citizens. We respectfully ask the EU to guarantee their rights unilaterally and as soon as possible, because not to do so may represent a violation of article 8 of the European Convention on Human Rights.
"We ask this also in the name of EU citizens resident in the UK who understand and empathise the distress suffered on a daily basis by their UK counterparts in the rest of the EU."
He told reporters he was encouraged by the stance adopted by EU in its commitment to secure the rights of five million EU citizens set to be most adversely impacted by the UK's departure from the EU.
"However, our evidence shows that the uncertainty these citizens are facing has reached a critical level. Their lives are in limbo and the current situation poses a real threat to the full enjoyment of their private and family life."
Dimitris Giannoulpoulos, a senior lecturer in law at Brunel University London, and the founder of the 'Britain in Europe' think tank, said, "Article 8 provides strong support to the argument that the EU must now unilaterally recognise the rights of UK citizens in the EU.
"Unless it does so, it risks causing irreparable damage upon the right to private and family life of 4.5 million Europeans, bringing disrepute to the system of human rights protection in Europe and overwhelming the administration of justice in affected EU countries."
Campaigners said the Conservatives' election manifesto and the failure to unilaterally guarantee the rights of EU nationals has caused further anxiety.
Ruvi Ziegler, of the University of Reading, told the press conference, "The pre-article 50 notification promise that rights of non-UK EU citizens will be a first priority for the UK, or the subject of an early agreement is gone.
"There is no way of reading the manifesto other than that non-UK EU citizens remain a 'bargaining chip', subject to the rest of the issues being resolved in a satisfactory manner.
"If 'no deal is better than a bad deal', as the Tory manifesto explicitly states, and if rights of non-UK EU citizens are one of the topics for negotiation, which other conclusion can be drawn?"
Ziegler added, "There is no compassion or recognition of the contribution that non-UK EU citizens make to British society.”
"This places the future life plans of five million EU citizens in jeopardy, that is to say 3.4 million EU citizens in the UK and 1.6 million British citizens in the EU."
New Europeans has called on the EU to be "even more bold and ambitious" than it has been up to now in its response.
Casale added, "In reality, what we are now facing is a human rights crisis on a mass scale inside the borders of the EU. European citizens look to the EU to continue to respond to this crisis in a way consistent with a vision of Europe as a Europe of the citizens and in a manner consistent with the EU's reputation as a guarantor of human rights."