Speaking at a public hearing on Thursday, the Belgian MEP also revealed he had received about 7000 messages from citizens concerned at how Brexit might affect them.
He told the hearing, "These are individuals who will be directly affected. In these thousands of emails and letters to me they are describing their personal situations and the common theme is that they do not know what is going to happen to them after Brexit. It is heartbreaking and this situation cannot go on."
Verhofstadt said the urgency of dealing with citizens' rights meant that Parliament, and not Council as has been proposed, should be the institution to first decide if sufficient progress has been made in the Brexit talks before the negotiations move forward.
Michel Barnier, the EU's chief negotiator in the talks, has said that sufficient progress must be made on the size of the financial settlement between the two sides before talks can move to other issues.
But Verhofstadt told the hearing, "Parliament should also be putting pressure on the Brexit negotiators to resolve the issue of citizens' rights before anything else is discussed."
Parliament, he suggested, should adopt a resolution on this issue, probably after the summer recess. This would be separate to the resolution on Brexit adopted on 5 April.
The ALDE group leader and other speakers stressed that citizens' interests must come first in the negotiations between the EU and the UK government with fair and equal treatment for the three million EU citizens living in the UK and 1.2 million British citizens living in the EU.
The hearing was organised by the civil liberties, employment and petitions committees.
Verhofstadt underlined the moral duty to end the uncertainty created for both EU citizens living in the UK and UK nationals in the EU since the 23 June referendum.
The EU, he argued, should let go of the principle that "nothing is agreed until everything is agreed" in negotiations, "because a quick solution for citizens' rights is a matter of priority."
Another keynote speaker, Anne-Laure Donskoy, of the '3 million' group set up to campaign for the Brexit rights of the three million EU citizens in the UK, said the EU and the UK government should negotiate a deal on the situation and rights of citizens as a matter of urgency and before starting the other Brexit talks.
She said, "Today, we are calling for the rights of these people to be ring fenced and for totally separate negotiations on this specific issue."
She added, "This is causing great anxiety for millions of people some of whom have told us they cannot open bank accounts or obtain mortgages because of the continuing uncertainty about their rights and status after the UK leaves the EU. Others have said they have been turned down for jobs or discouraged from even applying for the same reason."
Donskoy also condemned the registration process in the UK for those EU citizens who seek to apply for British citizenship.
Applicants have to complete an 85 page form and one third of applications, so far, have been rejected.
She said, "The system is creaking at the seams. The registration procedure is over complicated and unfair."
Other comments on the issue came from UK Socialist MEP Claude Moraes, Chair of the civil liberties committee, who said, "Today's hearing shows that Parliament stands ready to fulfil its role in the negotiations by working hard with our constituents, our home governments and the EU institutions to ensure the voices of concerned citizens are being heard."
Moraes added, "Human beings come first, we are not commodities and what happens on citizens' rights sets the tone for the entire negotiation and relationship for a generation to come, so we must get it right".
Romanian ALDE group member Renate Weber, a Vice-Chair of the employment committee, told the hearing, "I strongly believe that when thinking about Brexit consequences, there is no greater concern than the fate of EU citizens who study, work and settle in the UK and also visitors, and of those British citizens who work or live in EU 27."
She went on, "Brexit will have a direct impact on the lives of hundreds of thousands of workers and students, on millions of tourists, as citizens' mobility and rights are at the heart of the European project. We have worked hard to guarantee citizens' uninterrupted access to all kind of benefits, and these rights should be safeguarded."
Another speaker, Swedish ALDE group deputy Cecilia Wikström, who is Chair of the petitions committee, noted, "The Brexit decision has created uncertainty for three million citizens from other member states living in the UK, as well as for the 1.2 million British citizens currently living in the EU.
"These people are anxious and frightened about their future and their concerns must therefore be our top priority, and issues related to citizens' rights solved first of all."
She said, "We shall never forget that this concerns real people, they are not just pawns in the negotiations. This is about basic human values and about common decency. Only by putting the citizens first can we achieve a fair result in the end".