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International Trade Secretary Liam Fox said he "hoped" they would but it depended on whether other countries were "willing to put the work in".
He said more deals were coming, after signing one with Australia.
Concerns have been raised that the UK will leave the EU without a deal that would protect current arrangements. [...]
In 2017, Mr Fox said that the UK could "replicate the 40 free trade agreements before we leave the EU", so that there would be no disruption to trade.
But with just over two months to go until Brexit, not one has been signed, said the BBC's business correspondent Jonty Bloom.
The Department for International Trade says some agreements are at an advanced stage but none of the 40 free trade deals that the EU has with other countries have so far been rolled over so that they will cover the UK after Brexit.
The closest the UK has come to rolling over a free trade deal is an initial agreement with Switzerland to replicate the existing EU-Switzerland arrangements "as far as possible". But that deal has not been formally signed yet.
Asked about a report in the Financial Times that Britain would not be close to finalising most of the 40 free trade deals the EU currently has with other countries, Mr Fox told the BBC: "I hope they will be but there are not just dependent on the UK. Our side is ready.
"It's largely dependent on other whether countries believe that there will be no deal and are willing to put the work in to the preparations."
On Friday, he signed a "mutual recognition agreement" with the Australian high commissioner in London - to maintain all current relevant aspects of the agreement it has with the EU. The EU does not have a free trade agreement with Australia.
He said there would be a "pipeline of them to be signed as we go through" and the agreement made it easier for UK goods to comply with Australian standards.
Mr Fox also said that staying in a permanent customs arrangement with the EU would "not be delivering Brexit" as he did not believe it would allow the UK to pursue an independent trade policy. [...]