The current burgundy passport, in use since 1988, will revert to its original blue and gold colour from October 2019.
The boss of UK supplier De La Rue told the BBC that Franco-Dutch firm Gemalto had won the £490m contract.
Culture Secretary Matthew Hancock said a final decision had not been made.
However, the Home Office said passports did not have to be made in the UK and some blank covers were already made overseas.
De la Rue boss Martin Sutherland told the BBC's Today programme: "Over the last few months we have heard ministers happy to come on and talk about the new blue passport and the fact that it is an icon of British identity.
"But now this icon of British identity is going to be manufactured in France."
He added: "I'd like to ask Theresa May or Amber Rudd to come to my factory and explain to our dedicated workforce why this is a sensible decision to offshore the manufacture of a British icon."
Under EU procurement rules, the Home Office had been required to throw open the bidding process to European firms.
De La Rue, which has held the contract to manufacture British passports since 2009, said it it had been "undercut on price" by Gemalto.
Mr Sutherland said it was unclear whether jobs at the firm's Gateshead factory would be affected. De la Rue employed more than 600 staff at the site last year.
The company issued a profit warning on Tuesday, which triggered a sharp fall in its share price.
On Thursday morning, the shares were down by a further 5.7%.
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