Theresa May is facing a backlash after she said EU workers would no longer be able to "jump the queue" after Brexit.
SNP leader Nicola Sturgeon said the PM's words were "disgraceful" and EU citizens living in the UK said they would fuel hate crimes against them.
The comments came in a speech to business leaders on Monday, in which she vowed to end EU free movement after Brexit.
Downing Street said EU citizens made an "important contribution" to the UK.
The government says it wants the estimated 3.5 million citizens of other EU nations already living in the UK to stay after Brexit but they will have to apply for leave to remain.
But Mrs May used her speech to the CBI to highlight the fact that the proposed EU withdrawal agreement will bring an "end to free movement, once and for all".
The government has yet to set out details of what this will mean in practice in the longer term, but Mrs May has said at some point in the future, after the post-Brexit relationship between the UK and the EU kicks in, that EU citizens will be treated the same as those arriving from outside the EU.
"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," said Mrs May in her speech.
"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."
Scottish First Minister Nicola Sturgeon said the prime minister's language was "offensive" and "disgraceful".
"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."
The European Parliament's Brexit co-ordinator Guy Verhofstadt also criticised Mrs May.
"EU citizens living, working, contributing to UK communities, didn't 'jump the queue' and neither did UK nationals in Europe," he said.
"They were exercising rights which provided freedom and opportunities. We will fight to ensure these continue in the future, especially after any transition."
EU citizens living in the UK took to social media to accuse the prime minister of using them as scapegoats to shore up support for her Brexit agreement.
It's time we all woke up to the fact that @The3Million are being made scapegoats by a nasty, incompetent government, that lies and dog whistles instead of having the backbone to take responsibility for the mess they've created. Shame on you @theresa_may!https://t.co/0J2Imwi9LS— FinalSayForAll #FBPE (@FinalSayForAll) November 19, 2018
Citizens rights campaigner and professor of history at Northumbria University Tanja Bueltmann said the PM's "despicable" comments were a "punch in the face" for EU citizens like her who had made their home in the UK.
"We are your neighbours, colleagues, friends and family; your doctors, hairdressers and teachers," she said in a Huffington Post article.
Mrs May's words "essentially cast us as unwanted queue-jumpers, cheats and people without skills and talents undeserving to be here, in our own home," she added.
She accused the prime minister of trying to set one group against another - and said her words would mean EU citizens would have to "brace themselves for yet more hate over the coming months".
"The press called May's speech her Brexit fightback. It is not. But it is a fight on the back of EU citizens. In less than a week, May reduced us from 'valued citizens' to 'queue-jumpers.'"
Mike Galsworthy, a director of anti-Brexit campaign Scientists4EU, accused the prime minister of playing a "cynical" blame game to sell her Brexit agreement, in a video message on Twitter.
May just threw EU citizens in UK, expats & youth under the bus in order to sell her Brexit plan. It’s shameful. https://t.co/6J1PQvKeyI— Mike Galsworthy (@mikegalsworthy) November 19, 2018
"EU citizens already contributing to the UK didn't get here by jumping the queue, they got here by being the best person for that job," said Mr Galsworthy.
Any hurdles faced by workers from outside the EU, such as visa and financial stipulations, had been placed there by the UK government, he added.
The prime minister's official spokesman, when asked whether Mrs May regarded EU nationals currently working in the NHS and other public services as having "jumped the queue", 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.
"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."
The spokesman said an immigration white paper setting out the proposed new system was due to be published "shortly".