Asked about the time frame for such a deal, Truss told reporters: "I would definitely say months rather than years."
While Australia has taken years to negotiate each of its 15 bilateral and multi-lateral free trade deals, Birmingham was keen to fast-track a new pact with Australia's former colonial master.
"We stand absolutely ready to work with the U.K. at the moment they are ready to do so," Birmingham said.
Britain is giving priority to negotiating trade deals with the United States, Japan, Australia and New Zealand. Britain is also considering joining the Trans-Pacific Partnership multi-nation trade bloc.
Britain has been part of Australia's negotiation of a free trade deal with the E.U., which began in 2015.
In colonial times, Britain was Australia's most important trading partner. Distant Britain provided 70% of Australia's imports during the 19th century and bought 80% of Australia's exports. Britain remains Australia's biggest export market within the E.U.
Commonwealth countries including Australia and New Zealand felt abandoned in the 1970s when Britain joined the European Economic Community, the forerunner of the E.U.
Birmingham said the bilateral trade relationship before Britain joined the European trade bloc framed the history of the new trade discussions.
"If we held grudges over things that happened 40-plus years ago in relation to our trade or foreign policy or anywhere else, we'd never accomplish anything," Birmingham said.
Truss said Britain had been hampered from making trade deals outside the E.U. by increased regulation driven by Brussels.
"The U.K. trades more with the rest of the world than the E.U. and that is where a lot of the growing markets are," Truss said. "There is a big opportunity for us to do things differently and to use those new freedoms we have."