The Foreign Secretary said fox hunting was "part of the countryside" and that he was "happy for people to do it" despite it "not particularly (being) my thing".
However, Mr Hunt's push to change the law was condemned by the Labour Party, which said fox hunting was a "barbaric practice".
Mr Hunt said he would support a vote in Parliament when there was a majority in the Commons likely to back the move.
He said: "I don't hunt myself. It's not particularly my thing.
"Look, I think we have to recognise it's part of the countryside.
"And I think we have to recognise that in terms of the balance of the countryside it's part of our heritage.
"So personally I'm happy for people to do it."
The move comes as ballot papers for the Conservative leadership contest between Mr Hunt and Boris Johnson begin to be sent to the party membership.
Fox hunting was banned in England and Wales following the introduction of the Hunting Act 2004 - which came into force a year later.
The legislation permits drag hunting where hounds are trained to follow an artificial scent.
This is not the first time a potential repeal of the legislation has been suggested.
The Conservative-Liberal Democrat coalition agreement included a pledge to "bring forward a motion on a free vote enabling the House of Commons to express its view on the repeal of the Hunting Act which ultimately did not take place.
Theresa May also scrapped plans for a vote in 2018, saying: "I think there was a clear message about that and that's why I say there won't be a vote on fox hunting during this parliament."
The League Against Cruel Sports said the pledge showed Mr Hunt was "out of touch with public opinion".
They added: "The last time a politician said we should bring back hunting - Theresa May in the 2017 General Election - she was punished in the polling booths. Nothing has changed."
Ian Lavery, Labour Party chairman, said: "This Tory leadership race is going from bad to worse.
"We've had Johnson's tax cuts for the wealthy, a race to the bottom on no-deal Brexit, and now a pledge to bring back this barbaric practice that Labour had proudly banned."