Theresa May could be accused of racism in the wake of the Windrush immigration scandal, a Labour frontbencher has said.

Shadow women and equalities secretary Dawn Butler said the Prime Minister’s policies were delivering “institutionalised racism”.

Asked if Mrs May could be accused of racism, Ms Butler told Sky News: “Yes, she is the leader. She is presiding over legislation … discriminating against a whole group of people who came from the Commonwealth, who suffered racism when they came over – the ‘no blacks, no Irish, no dogs’ – and now they are having to relive that trauma all over again because of Theresa May.”

Pressed on whether Mrs May was racist, the frontbencher said: “In my own personal opinion, and I’m speaking as myself as Dawn Butler, the daughter of Jamaican parents, I’m saying that Theresa May has presided over racist legislation that has discriminated against a whole generation of people from the Commonwealth.

“Her policies, that she has implemented, have disproportionately affected people from the Commonwealth and people of colour.

“Therefore, if you look at what institutional racism is, that’s what her policies are currently delivering.

“So, Theresa May has to not only reconsider her position, but she has to reconsider her policies, and an apology is not good enough.”

It comes after a slew of stories about members of the Windrush generation facing problems with access to healthcare and other state services after the “hostile environment” policy intended to clampdown on illegal immigrants was brought in by Mrs May in 2014.

Many in the Windrush generation – who arrived from the Caribbean between the late 1940s and 1970s – have no record of their status and have found it challenged under recent laws that require them to provide proof of near-continuous residence.

The controversy has seen Home Secretary Amber Rudd face fresh calls to quit.

Labour’s shadow foreign secretary Emily Thornberry, speaking on BBC One’s Andrew Marr Show, said there was something “rotten at the heart of Government” as she slammed the Home Office’s approach to immigration.

She said: “People have died, people have lost their jobs, lost their futures. People working in the national health service all their lives suddenly lose their jobs.

“It could not be worse and yet the Home Secretary thinks ‘I can apologise and it will be alright’. Well, it won’t be. I really think she should quit.”

When asked whether she believed ministers had implemented a racist policy, Ms Thornberry said: “I don’t like to bandy around these things, I’d rather just stick to the evidence.

“I’m happy to say there is something rotten at the heart of Government.”

Justice Secretary David Gauke said the Home Secretary should not resign, telling the BBC: “No, because when it comes down to it, the central policy is right.

“Clearly, there have been very significant failures in terms of how this has been implemented.

“I think it is right that both the Home Secretary and Prime Minister have apologised for this.”

Asked if he felt ashamed about what had happened, Mr Gauke said: “Yes. It is wrong what has happened. It should not have happened.”