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Theresa May must adopt an Australian points-based immigration system by 2020 or face a backlash from Brexit voters, two former cabinet ministers have said.

Theresa Villiers, the outgoing Northern Ireland secretary, said in an interview with The Telegraph that the country voted to end the “free for all” of “open door” immigration at the referendum.

Iain Duncan Smith, the former work and pensions secretary, also said that people are “living in cloud cuckoo land” if they think voters did not choose complete control of their borders by backing Brexit.

The comments will add pressure on the Prime Minister to adopt a hard line in negotiations over Brexit and exposes a growing cabinet split on the issue.

At least five cabinet ministers are believed to support introducing a points-based system: Liam Fox, Boris Johnson, Priti Patel, Chris Grayling and Andrea Leadsom.

However Amber Rudd, the new Home Secretary, criticised the idea during the EU referendum campaign and neither Mrs May or Phillip Hammond supported the move earlier this year.

Sources have suggested that the government will not take a stance on the idea until the end of Brexit negotiations, which is at least two years away.

Before the referendum Tory Brexit campaigners pledged to introduce the system, which allows a country to control how many migrants enter based on the need for skills in certain industries.

Talking to The Telegraph, Mrs Villiers said matching migrant numbers to “skills shortages” was a “sensible way to have a reformed immigration system”.

Asked if it should be in place by 2020, she said: “That’s an achievable and sensible goal. I very much hope that it’s going to be possible to deliver that.”

Mr Duncan Smith said he also believed in a “points and permits” system, adding: “It is my view that we will have left the EU by the time of the next election which means we have to have a system in place that controls migration.

“People are living in cloud cuckoo land if they think the British voters did anything else but absolutely vote to take back control. It was very explicit that means migration and that’s exactly what they voted for.”

Speaking for the first time since her cabinet sacking last month, Mrs Villiers also reveals her “loss” and “grief” at leaving office and says she fears “this is the end” of her ministerial career aged just 48.

“It was a difficult conversation because I was obviously pretty upset, to be honest,” Mrs Villiers says of her meeting with Mrs May. “These kind of conversations are never easy.”

She added: “Politics is an uncertain path for anyone. It certainly comes with its ups and downs. To be very frank, I suppose I’m still suffering regret, grief, loss as you do in these kinds of situations. But I’ve just got to accept it and move on.”

“I’m really sad to have left my Northern Ireland role behind. It’s a very special place, I left a lot of friends, the people are great and I felt I was able to achieve things.” She named helping solve the political stand-off at Stormont last year among them.

Mrs Villiers also insisted passport controls are not needed on the Northern Ireland border, rejected calls for an early election and defended Mr Cameron’s controversial honors list.