An Australian-style immigration system was one of the key policy pledges made by Vote Leave campaigners including Boris Johnson during the referendum campaign, reports the Press Association.
However, May said there were questions about whether such systems ever worked, but vowed that free movement of European Union citizens could not continue in its present state after Brexit.
He said: “Theresa May’s track record on immigration as Home Secretary was appalling and her comments rejecting an Australian-style points system really worry me.
“There is already huge anxiety out there in the country regarding Theresa May’s reluctance to trigger Article 50. Her rejection of the type of migration system so many went out there and voted Leave to see implemented indicates serious backsliding.
“Those of us on the Leave side were perfectly clear in campaigning for strong border controls outside of the EU.
“The people were clear in wanting a points-based immigration system which is why so many went out and voted to Leave the European Union. Any watering down from that will lead to real anger.
“Given that myself and others also campaigned for a migration system that would treat all who wanted to come equally, any preference for EU nationals would be totally unacceptable
“If the establishment think they can stitch-up Brexit then they better be ready for the huge electoral consequences from a British public who on June 23rd voted for radical political change and now expect it to be delivered without failure.”
The Prime Minister also refused to rule out contributing funds to EU programmes after Brexit, despite the money paid to Brussels being another major issue in the referendum debate.
May would also not guarantee to meet the Vote Leave pledges of £100 million a week for the NHS or a cut in VAT on energy bills.
As home secretary, May spent six years struggling but ultimately failing to reduce net migration to below 100,000 a year.
Speaking to reporters during her visit to the G20 summit in China she said: “You really don’t want to ask a former home secretary about the intricacies of points-based systems. It might take a very long time answering your question.
“One of the issues is whether or not points-based systems do work, and so forth.
“What I say is the voice of the British people was very clear. They wanted control in the issue of the movement of people coming in from the European Union.
“They didn’t want free movement to continue as it has done in the past. We will be going out there to deliver on that.”
She added: “A lot of people talk about the points-based system as always being the answer in immigration. There is no single silver bullet that is the answer in terms of dealing with immigration.
“You have to look across the board, you have to look at the whole range of issues, not just how you bring control through the rules we have for people coming in, but also making sure you are rooting out abuse in the system and obviously dealing with people if they are discovered here illegally.”
Challenged that people thought a points-based system was what they were going to get from Brexit, May said: “People voted for control. What they wanted to see was some control in the movement of people from EU countries into the UK.”
Asked if she would rule out contributing funds to the EU she said: “What we are doing at the moment is looking, making our preparations before we actually trigger Article 50 and get into the formal negotiating process.
“I think what we need to be doing is making sure that we can get the best possible deal for the UK when we leave the EU.
“I’m optimistic about that, I think we should be ambitious about that, and that’s what I am.
“But I’m not going to give away my negotiating hand. I want to actually, let’s do that work, let’s be ambitious about what we can achieve and let’s go out there and work for it.”
She also repeated her desire for EU nationals to be allowed to remain in the UK after Brexit - but only if the rights of Britons overseas are respected, indicating it would be a priority in the talks with the other 27 EU members.
“I expect to be able to guarantee their status, I want to be able to guarantee their status. The only circumstances in which that wouldn’t be possible is that if other EU countries don’t guarantee the status of British citizens who are living in other EU countries.
“It’s right and proper that I, as Prime Minister, say I must have consideration for British citizens who are living in the rest of the EU.
“I hope this is an issue we can resolve at an early stage.”
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