Theresa May described the move as "very encouraging" and insisted it showed Brexit can work for Britain.
In a phone call to the new PM, her Australian counterpart Malcolm Turnbull said he urgently wanted to open up trading between the two countries.
Liam Fox, the new International Trade Secretary, said he was already "scoping about a dozen free trade deals".
But Britain cannot sign any deals while it is still an EU member.
Mrs May said: "I have been very clear that this government will make a success of our exit from the European Union.
"One of the ways we will do this is by embracing the opportunities to strike free trade deals with our partners across the globe. It is very encouraging that one of our closest international partners is already seeking to establish just such a deal."
"This shows that we can make Brexit work for Britain," she added.
Dr Fox, a prominent Brexit campaigner, said that numerous non-EU countries had already asked Britain for a trade deal and he was "scoping about a dozen... to be ready for when we leave".
It comes amid reports he is preparing to fly to the United States next week.
He told the Sunday Times: "We`ve already had a number of countries saying, `We`d love to do a trade deal with the world`s fifth biggest economy without having to deal with the other 27 members of the EU`."
UK offered Brexit free trade deal with Australia
17 July 2016
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Australia has called for a free trade deal with Britain following its exit from the European Union.
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