In a letter to the president of Guernsey’s Policy and Resources Committee, Deputy Gavin St Pier, Mr Johnson said that the UK and the Dependencies ­had become closer during Brexit preparations.

The correspondence, which was made public yesterday, came in response to a letter from Deputy St Pier sent in July.

‘I want to be clear that the UK government appreciates the special relationship that the UK has with the Crown Dependencies,’ wrote the Prime Minister.

‘I am keen to further strengthen this relationship after the UK has left the European Union, when the UK government will be negotiating its own free-trade agreements on behalf of the entire British family, including the Crown Dependencies.

‘The UK government will continue to engage the Crown Dependencies’ governments throughout this process as we seek a future relationship with the UK that works for the entire British family.’

Mr Johnson also acknowledged the islands’ commitment to eventually publish public registers of beneficial ownership – effectively a list of who ultimately controls or owns assets held by companies registered in the jurisdictions.

This year the Dependencies have faced mounting pressure to increase corporate transparency after two UK MPs attempted to bring forward legislation which would have forced the islands to publish public registers.

The proposals were withdrawn, but later in the summer the Dependencies issued a joint statement announcing they would produce public registers when it becomes the international standard.

Mr Johnson wrote: ‘The UK government is unambiguous in supporting the constitutional position that the UK Parliament should not legislate on your behalf on these matters without your consent.’