British businesses could be left up to £20 billion worse off if the customs plan favoured by Brexiteers is implemented, the head of HM Revenue and Customs has warned.

The so-called "max fac" plan to use modern technology to solve the Irish border question would leave firms facing huge charges for customs declarations and for EU "rules of origin", Jon Thompson told MPs.

In contrast the new customs partnership believed to be favoured by British Prime Minister Theresa May would cost a maximum of £3.4bn and may end up having "a net cost of zero or less", he told the Treasury Committee yesterday.

The two plans are at the heart of the British government's current impasse over what happens to the Irish border after the UK quits the European Union in 2019.

Mr Thompson, HMRC's chief executive and permanent secretary, told the committee customs declarations would cost an average of £13bn under max fac, with billions more to be added for EU rules of origin checks - like making sure cheddar cheese comes from Cheddar.

He told MPs: "I think you need to think about the highly streamlined customs arrangement (max fac) costing businesses somewhere in the late teens of billions of pounds to operate.

"Somewhere between £17bn and £20bn, it is that sort of order."

Max fac is the model favoured by Brexiteers including Foreign Secretary Boris Johnson and Environment Secretary Michael Gove.

It would use trusted trader arrangements and technology like number plate recognition cameras to avoid the need for border checks.

The customs partnership model, in contrast, would see the UK collects tariffs on behalf of the EU for goods intended for the bloc, with traders potentially able to claim a rebate if British duties vary.

Mr Thompson told the committee that because the customs partnership tried to deliver a "free-flow of goods" it avoided the costs linked to max fac.

While the maximum cost could be £3.4bn, he added: "The businesses would be getting that back on the tariff, so it might spend £100 and get £150 back.

"You could argue that the NCP has a net cost of zero or less."

Mr Thompson has earlier hit the max fac plan with another blow, with another committee saying he had warned it could take up to three years to introduce.

Hilary Benn, chairman of the Commons' Exiting the European Union Committee, said Mr Thompson had written to them saying that some elements of the max fac plan "could take around three years, which takes us beyond December 2020", the end of the proposed transition period.

Downing Street refused to be drawn on "speculation" about the implementation of a future customs model before the Cabinet had made its decisions.

The prime minister's official spokesman said: "The Prime Minister has asked for work to be done on both customs models, that work is ongoing and therefore any speculation about implementation is just that."

The spokesman said the £20bn cost was "not a figure that I'm aware of".

Asked if the prime minister was confident a new customs model could be in place by the end of 2020, the spokesman said: "We have been clear on a number of occasions that our intention is to be ready for the end of the implementation period."

A fallback option is set to be put forward by the UK if no alternative solution is found to address the problems caused by Brexit at the border with Ireland.

There is unease among Brexiteers about the proposed "backstop" arrangement to keep the UK closely tied to EU customs rules if alternative arrangements are not agreed with Brussels or able to be implemented.

The Foreign Secretary left no doubt he wants the UK to quit the customs union as soon as possible, telling Bloomberg TV during a visit to Argentina: "The Prime Minister is the custodian of the plan, which is to come out of the customs union, out of the single market and to get on with it, to get on with that project with all convenient speed, and that is what we are going to do."