The offer, made over dinner on Thursday during a summit of EU leaders in Brussels, was described by May as "fair and very serious."
Many EU citizens in the UK, and Britons living abroad, are worried about their status once Brexit happens. The estimated 1.3 million Britons living in Europe are similarly concerned about their post-Brexit legal rights. The UK's exit deadline is 30 March 2019.
Addressing other EU leaders at her first summit since the general election, the UK Prime Minister said she did not want anyone to have to leave or families to split up. "No one will face a cliff edge," she said.
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A new 'UK settled status' would grant EU migrants who had lived in the UK for five years rights to stay and access health, education and other benefits.
The proposals, which come on the first anniversary of the EU referendum, are dependent on EU states guaranteeing Britons the same rights.
The EU says that the cut of date for EU citizens arriving in the UK to get settled status must be March 2019 when the UK is due to leave the EU. May spoke further about the offer when she arrived at the European Council on Friday.
Asked by reporters if she thought there was any realistic chance of "getting anything better" from the EU side, May said, "Well last night I was pleased to be able to set out what is a very fair and a very serious offer for EU citizens who are living in the UK and the government will set out more detailed proposals on Monday.
"I want to reassure all those EU citizens who are in the UK, who've made their lives and homes in the UK that no one will have to leave, we won't be seeing families split apart."
She went on, "This is a fair and serious offer. And I want to give those EU citizens in the UK certainty about the future of their lives but I also want to see that certainty given to UK citizens who are living in the European Union.
"Of course there will be details of this arrangement which will be part of the negotiating process but we've made what I believe is a very serious, a very fair offer which will give reassurance and confidence to EU citizens living in the UK about their future."
May was also asked whether she thought the UK had conceded too much on the rights issue and, if so, if that meant the Brexit talks, which only formally started on Monday, were going well for the EU.
On this, she replied, "There was a very constructive start to the talks at the beginning of the week. We've set out the issues that we want to start talking about. I've said right from the beginning that I want citizens' rights to be one of those early negotiations and it will be.
"We've set out what I believe is a serious offer, a fair offer that will give the reassurance to EU citizens living in the UK, they've made their homes, they've made their lives in the UK and they will be able to stay and continue to do so."
Denis MacShane, a former UK Minister for Europe, said, "May's disastrous decision to abolish identity cards in 2010 means the UK has no idea of the number of fellow Europeans who live and work among us nor names of dependants. Many work here but have families back home."
The former Labour MP added, "It is going to be a giant bureaucratic nightmare to list all these people and deciding if children or elderly parents can live here as a family unit. We will then have first and second class categories of EU citizens."
MacShane went on, "The same is true of Brits in Europe. In Spain, only 300,000 are officially registered, but UK embassy officials in Madrid reckon there may be up to one million Brits with a home or stake in Spain. What will their rights be?
"The dream of retiring to warm southern Europe will now end because if the UK enacts discriminatory immigration measures against EU citizens after Brexit we can be sure there will be reciprocal action against Brits.
"It is all the more odd as according to the Oxford University Migration Observatory the number of east Europeans coming to work in the UK shrunk in the first quarter if the year to 26,000 compared to 40,000 in the same period last year and far below the 110,000 who arrived in 2007.
Further reaction came from the former British Liberal MEP Andrew Duff, who was also scathing of the UK citizens' rights offer, telling this website, "I think we knew all this beforehand. I hope she hasn't wasted her trip."
Looking ahead, Open Europe's Director Henry Newman said, "The Brexit path will likely still be towards an arrangement out of the single market and the customs union, with a full and comprehensive trade and customs deal."