"That's the date I'm working to, which could be brought forward if necessary," International Trade Secretary Liam Fox told the Sunday Times newspaper.
This would involve triggering Article 50 of the EU's Lisbon Treaty, which would start a two-year clock running on Britain's exit from the bloc, by the end of this year.
Before taking office last week following the June 23 vote for Britain to leave the EU, Prime Minister Theresa May indicated she would invoke Article 50 early next year.
Fox revealed that he had opened "very fruitful" trade talks with Canada on Friday, and was reported to be heading to the United States next week.
May also discussed a trade agreement with Australia in a phone call with Prime Minister Malcolm Turnbull on Saturday, her Downing Street office said.
Turnbull had expressed his desire for a free trade deal as soon as possible, the statement said.
May had campaigned for Britain to stay in the EU in the referendum but says she will make Brexit "a success".
"One of the ways we will do this is by embracing the opportunities to strike free trade deals with our partners across the globe," she said in the statement.
"It is very encouraging that one of our closest international partners is already seeking to establish just such a deal."
Britain cannot sign trade deals until it legally leaves the EU, but Fox said he would line up agreements for the day Brexit comes into force.
EU leaders are pressing for a quick divorce, but May has urged them to give her time.
One point of delay may be Scotland, which voted to stay in the EU while England and Wales voted to leave, and is threatening independence as a way of staying in the bloc.
After talks on Friday with Scottish First Minister Nicola Sturgeon, the leader of the secessionist Scottish National Party, May said she wanted a common British position.
"I won't be triggering Article 50 until I think that we have a UK approach and objectives," May told reporters.
Sturgeon told the BBC on Sunday that this "puts Scotland now in a very, very strong position" to influence events.