Mr Verhofstadt has warned the European Parliament will not support dilution of its citizens’ rights in order to move Brexit negotiations onto “phase two”, which includes trade and other post-divorce arrangements.
While the EU Parliament is not directly involved in Brexit negotiations, it has the final say over the final deal before it can go into effect.
Mr Verhoftstadt said: “My concern is that citizens’ rights are not being well-managed.
“I fear this problem will not be solved in the coming weeks.”
Negotiators for the UK and Brussels are keen to progress talks into the second phase, but that can only happen when EU leaders declare “sufficient progress” has been made in three key areas.
The areas include the financial settlement – the “divorce bill” – the border dividing Northern Ireland from the Republic of Ireland and citizens’ rights.
Failure to move forward on those issues would create further delays to the Brexit timetable.
A decision is expected at the next European summit in mid-December.
Mr Verhoftstadt said the Parliament is concerned Europeans living in the UK will be forced to navigate a complicated bureaucratic maze in order to maintain their residency rights.
He said it is also not clear what rights family members of Europeans living in the UK will enjoy.
The Parliament identified the issue as one of its “red lines” for the talks and has been urging action for weeks.
He added: “If this is not done in the right way, millions of people will be affected.”
European Commission President Jean-Claude Juncker said: “I don’t even understand this problem. Why not say, easily, with common sense … that things will stay as they are?
“The Europeans, ‘foreigners,’ as they are saying in London, they are there on the island, and so many British friends are here. Let them here, let them there. Why are we discussing nonsense like that?”
There are more than three million EU citizens living in the UK and more than one million British living in the EU.
Mr Verhoftstadt’s recent comments come after he claimed Britons can still become the “victims” of the historic referendum.
Mr Verhofstadt called for a “declaration” system for EU citizens living in the UK, to avoid them having to make an “application” to remain in the country after Brexit.
He said declarations should cover a “whole household” rather than an individual and said the “burden of proof” must be on the Home Office rather than the household.
Mr Verhofstadt added: “We cannot accept a system where EU citizens living in the UK, or are mainly already permanent residents, are at risk of being deported as a result of the process.”
The Government issued a technical paper detailing a two-year grace period for EU nationals to apply for settled status once the UK leaves the bloc.
Brexit Secretary David Davis said the UK “will support everyone wishing to stay to gain settled status”.