Amid ongoing anger over the Windrush scandal, Lib Dem Lords leader Lord Newby, who was party chief whip in the Upper House during the coalition years, told The Yorkshire Post the ex-Prime Minister Mr Cameron asked his deputy Sir Nick to intervene on immigration "because he couldn't stop" Mrs May.
His comments came after former civil service head Lord Kerslake said coalition ministers were so unhappy with the immigration policies brought in by Mrs May when she was home secretary they made comparisons with Nazi Germany.
Labour has said Mrs May's "hostile environment" tightening of immigration rules in 2014 was directly responsible for Commonwealth citizens who were invited to Britain to rebuild the country in the decades following the Second World War being wrongly threatened with deportation.
Lord Newby claimed that during the coalition years Mr Cameron and his Chancellor George Osborne were opposed to some of Mrs May's more hardline immigration policies.
But The Yorkshire Post understands it was often Sir Nick who had to negotiate with Mrs May over controversial immigration proposals.
In an interview with this newspaper, Lord Newby said: “The phrase Nazi Germany isn’t one I remember.
“But you remember those appalling ("Go Home") vans that we had to put a stop to.
“And more generally during the coalition it was Theresa May not just against the Lib Dems but Theresa May against her own Prime Minister and Chancellor.
“And Nick Clegg used to be sent round by David Cameron to intercede with Theresa May to stop her being so extreme on immigration because he couldn’t stop her.”
Lord Newby, who grew up in Rothwell in West Yorkshire, claimed the entire Cabinet was agreed that students should not count towards the Tories' target of reducing net migration to the "tens of thousands", but Mrs May refused.
“Everybody, bar her, wanted to exclude them but she wouldn’t, she’s very, very stubborn," he said.
In an interview with BBC2's Newsnight on Wednesday, Lord Kerslake said the 2014 Immigration Act had been a "very contested piece of legislation" across Whitehall.
"Now I can't say, and shouldn't say, as the former head of the civil service, precisely who gave what advice to whom. But what I can tell you, it was highly contested and there were some who saw it, I shan't name them, as almost reminiscent of Nazi Germany in the way it's working," he said.
Asked if he was referring to people in the civil service, Lord Kerslake said: "No, some in the ministers were deeply unhappy."
Environment Secretary Michael Gove, who was a minister in the coalition, flatly rejected the comparison with Nazi Germany.
He told BBC Radio 4's Today programme: "I have never heard anyone make that comparison before Lord Kerslake did.
"It is not for me to criticise a distinguished former public servant like Lord Kerslake, but I respectfully disagree."
Downing Street defended the changes, saying they had been thoroughly debated by MPs at the time.
"The changes were fully debated, fully scrutinised by Parliament. It is important that we have an immigration system which is robust and in which people can have faith," the Prime Minister's official spokesman said.
"Where there are issues in relation to the Windrush generation not having received sufficient support in order to establish their status, she (Mrs May) has apologised and the Home Secretary has apologised."
Home Secretary Amber Rudd has said no one who is entitled to be in Britain will be removed and promised the Government will do all it can to those affected - many of whom came to Britain as schoolchildren in the 1950s and 1960s - to resolve their immigration status.
Labour leader Jeremy Corbyn said the Government needed to act immediately to deal with the issue.
Asked if Ms Rudd should consider her position, he told the Press Association: "I want those responsible to deal with the problem straight away and if they can't, move on and let somebody else do it."