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The chief executive of HM Revenue and Customs told MPs firms would have to pay £32.50 for each customs declaration under the so-called "max fac" solution.
Jon Thompson said any new system could take up to five years to fully work.
No 10 said the £20bn figure was "speculation" but "issues" remained with both options being considered.
The figure is higher than the £13bn UK contribution to the EU in 2016.
Leading Tory Brexiteer John Redwood said he did not accept what he described as the HMRC's "general figures" but told the BBC that "if it is going to cost this much it is the wrong system". [...]