Dr Fox made the comments after visiting Japan to represent the UK in a meeting of G20 trade ministers.
Mr Farage accused the Government of being “simply not prepared” to hold trade talks with the US earlier this week.
He claimed he was gathering a “team of industrialists and dealmakers” who would then visit Washington “on a trade mission and get this thing moving”.
The criticism came after last week’s state visit by Donald Trump, when Mr Farage claimed the UK was “behind the curve” on plans for a post-Brexit trade deal with the US.
In a series of tweets, Dr Fox stated “only the UK Govt speaks for our country internationally” in a direct riposte to Mr Farage’s attempt to muscle his way into negotiations.
Rejecting the former Ukip leader's assertion the UK did not have “competent” trade negotiators, Dr Fox insisted he had a “large team of expert staff”.
But he also flagged up that they would be ready to begin negotiations once the UK had left the EU and not before, which is not allowed under international law.
He stressed those talks on a large-scale trade deal would not only be with the US but also Australia and New Zealand.
The International Trade Secretary also accused Mr Farage of “misleading” people over what the UK was able to do while still a member of the EU.
He tweeted: “It is entirely misleading to suggest that we could already be negotiating trade agreements. You know full well that we cannot begin formal negotiations until we have left the EU. We have been laying the groundwork for UK/US FTA (free trade agreement), including through our joint UK/US working group.”
Dr Fox issued the rebuttal after visiting Tsukuba in Japan to meet international counterparts.
Following the talks, he called for urgent action to protect the World Trade Organisation (WTO) from an “existential crisis”.
But Dr Fox, who is also president of the Board of Trade, said the agreement reached did not address “the scale nor urgency” of the challenges and urged partners to go further.
He said: “The message from the UK is clear: urgent and drastic action must be taken to protect the global trading system as we know it.”
Earlier this week, Boris Johnson’s bid to succeed Theresa May as prime minister – and pick up the Brexit baton – was bolstered by a resounding victory in the Tory leadership contest ballot of Conservative MPs.
The former foreign secretary, who has put delivering Brexit at the heart of his pitch for Number 10, opened up a considerable lead, pulling 71 votes ahead of his six rivals.
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