The prime minister said changes after Brexit mean EU citizens will no longer be given an advantage over people coming from the rest of the world.
Her comments, made in a speech at the CBI conference, were attacked by Scottish first minister Nicola Sturgeon.
Calling the remark “offensive” and “disgraceful”, the SNP leader said: "That the case for Brexit has been reduced to such a miserable and self-defeating bottom line is depressing in the extreme. Let's lift our sights higher than this.
"Actually, the more I think about it, the more offensive 'jump the queue' is as a description of a reciprocal right of free movement. Really disgraceful."
And writing for HuffPo, Tanja Bueltmann, pro vice-chancellor and a history professor at Northumbria University, said: "May’s words today are not only insulting, but also directly harmful. They continue to invoke the lies that EU citizens here have not contributed.
"It is in such rhetoric that we can find the roots of hate against us. This hate can manifest in verbal abuse – from telling us to ‘f*** off back to the s***hole’ we came from to labelling us ‘enemy aliens’ – but it is not its only form.
"Home Office figures clearly show a rise in hate crimes against EU citizens, and police already predict another rise as Brexit day comes closer. The EU referendum, and everything May and her government have done since then, directly enabled this."
Accusing Ms May of "pinning people against people, casting one group as better than another", she added: "That is despicable. And it is one of the hallmarks of xenophobia. As a result, there can now be no doubt that EU citizens need to brace themselves for yet more hate over the coming months."
Downing Street said the prime minister recognised the “important contribution” made by EU migrants in the UK.Ms Sturgeon will meet Ms May on Wednesday evening to discuss Brexit.
The government has put forward proposals for a new immigration system that will come into effect once the Brexit transition period ends.
It will prioritise granting visas to workers with skills the UK needs, and will treat EU citizens the same as people from non-EU countries.
Ms May also faced criticism from Guy Verhofstadt, the European Parliament’s Brexit co-ordinator.
He said: "EU citizens living, working, contributing to UK communities, didn't 'jump the queue' and neither did UK nationals in Europe.
"They were exercising rights which provided freedom and opportunities. We will fight to ensure these continue in the future, especially after any transition."
The 3.5 million EU citizens living in the UK will have to apply for leave to remain in the country after Brexit, although the government has said it wants them to stay.
However, after the transition period, EU nationals will not be able to move freely to the UK, and will instead have to apply for a visa along with everyone else.
Of the plan, Ms May told the CBI: "It will no longer be the case that EU nationals, regardless of the skills or experience they have to offer, can jump the queue ahead of engineers from Sydney or software developers from Delhi.
"Instead of a system based on where a person is from, we will have one that is built around the talents and skills a person has to offer."
Groups representing EU citizens in the UK condemned the prime minister’s comments.
Writing in the Evening Standard, Maike Bohn of The 3 Million said: “Theresa May's comment at the CBI conference that her Brexit deal will stop EU citizens 'jumping the queue' is wrong on every level - morally factually and economically.
“Mrs May paints a picture of faceless EU citizens preventing highly skilled fellow Commonwealth citizens from, say, India and Australia working in the UK but the rigid regulations created by Mrs May herself make it hard for Commonwealth citizens to work in the UK.
“The government is cranking up the anti-EU citizens rhetoric yet again to sell its Brexit deal. I appeal to the British public not to buy it."
Asked about the backlash, Ms May’s official spokesman said: "We have always been clear of the important contribution which EU citizens make to our economy and to public services.
"The point the prime minister is making is that we wish to have a global system where people's skills are the basis on which they are able to work in the UK.
He added: "At the moment, we have two systems in place - one for people coming to the UK from outside the EU and one for people coming under the rules of free movement.
"It's a fact that people coming under the system of free movement don't need a visa and those from outside the EU do."
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