The civil service will not be able to implement Brexit while carrying out its other duties after having been shrunk to its smallest size since the second world war, Bob Kerslake, a former Whitehall chief, has warned.

The former head of the civil service under David Cameron added to recent warnings that Whitehall was not sufficiently equipped to cope with the extra demands of leaving the EU.

It comes after a note for the Cabinet Office from Deloitte, a professional services firm, was published last week suggesting an extra 30,000 people could have to be hired to deal with the additional work.

Lord Kerslake, who led Whitehall from 2012 to 2014, told the Commons public administration committee that the government should start an immediate review of civil service numbers and capacities.

Asked about carrying out Brexit, he said: “It’s not possible to do that at a point when the civil service is at its lowest numbers since the second world war and continuing to fall.

“I think it is pretty essential for the government to pause, review, take stock of what it has in front of it and then revisit the question of capacity.

“And I don’t think it’s necessarily about lots of skills that the civil service doesn’t have – I think it has huge skills but there will be a real issue about numbers to deliver these very big demands.”

He advised there was a real issue about Whitehall’s capacity to manage the “huge, complex and big stakes” process of Brexit while “at same time taking forward a set of other policy initiatives”.

Other warnings about the scale of the challenge of leaving the EU have been delivered by Gus O’Donnell, a former cabinet secretary, John Manzoni, the Cabinet Office permanent secretary, and Simon Fraser, a former Foreign Office permanent secretary.

Asked if the civil service was prepared, Lord O’Donnell told The House magazine last week: “There’s a very simple, short answer to that, which is no. Brexit imposes a lot of extra requirements on the civil service. They’re not perfectly ready.

“Are they capable and in the process of gearing themselves up for it? Absolutely. I feel very confident that they will get there, but it will mean bringing in new people, developing the skills in all sorts of areas and expanding them into other areas.

“I’m confident that they will get there. But no one should be under illusions – this is an enormous job.”

However, Robin Butler, who led the civil service for 10 years from 1988 to 1998, was more optimistic, saying it was a “terrific opportunity” for officials.

“I haven’t often, since I retired, wanted to be back in the civil service, but I do now because I think that this is a very exciting time,” he told the public administration committee.

“I think there’s an opportunity to rise to it and I am confident that on past form the civil service will. Ministers are going to have to rely on civil servants during this process much more than they have been relying on them in the past – that is a terrific opportunity.”