The UK has yet to finalise agreements to replace existing free trade deals the EU has with 40 big economies if there is a no-deal Brexit.
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.
The UK is due to leave the EU on 29 March, under the Article 50 process and the UK’s EU Withdrawal Act, with or without a deal – unless the UK chooses to revoke Article 50 and continues as a member of the EU.
MPs defeated the withdrawal deal negotiated with the EU by a huge margin earlier this week, which provided for a “transition period” of 21 months, under which much of the UK’s relationship with the EU would remain the same.
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.
Some opposition parties have been making the case for a customs union. Theresa May held talks with the leaders of parties including the SNP and the Lib Dems, about a way forward after she won a confidence vote by a narrow margin in the Commons on Wednesday.
She also spoke to German Chancellor Angela Merkel and Dutch Prime Minister Mark Rutte on the telephone on Thursday night, and will be speaking to more EU leaders over the weekend.
But Labour Leader Jeremy Corbyn, who wants the UK to be in a permanent customs union with “strong” ties to the single market, has refused to take part in talks with the prime minister until she rules out the prospect of leaving the EU without a deal.
In a letter to Mrs May, Mr Corbyn said her talks were “not genuine”. He also accused her of “sticking rigidly” to her withdrawal agreement.
As many as 20 Tory ministers have also said they would quit the government unless the prime minister allows them to try to stop a no deal Brexit, according to the Telegraph.
Mrs May says ruling out no deal is impossible as it is not within the government’s power.
Writing in the Financial Times, Shadow Chancellor John McDonnell said the Conservative Party was “riven with division” so Labour would “return to Parliament to promote the compromise we believe is not only in the best interests of our economy but is also capable of securing sufficient support both here and in Brussels”.
If Parliament was at an impasse, and Labour could not get a general election “we should also retain the option of seeking a public vote,” he added.
Mr Corbyn has come under pressure from dozens of his MPs to back calls for another EU referendum. On Friday a pro-referendum campaign group paid for a wrap-around advertisement in his local newspaper, the Islington Tribune , urging him to back a “public vote on Brexit”.
In a speech at JCB Headquarters in Rocester, Staffordshire, former foreign secretary Boris Johnson said changing the date of leaving from 29 March would be “shameful”, and the public would view it as “an elite conspiracy to thwart Brexit”.
He instead urged the government to use Brexit to “unite the country”.