Tory leadership contender Jeremy Hunt has offered MPs a free vote on lifting the ban on fox hunting in England and Wales if he becomes prime minister.
But the foreign secretary said he thought the law was "not going to change" and declined to say whether he considered the activity "cruel".
Meanwhile, leadership rival Boris Johnson has promised to recruit 20,000 more police over three years.
The pledges come as Tory members begin to choose Theresa May's replacement.
Voting closes on 22 July, with the new leader - and prime minister - expected to be announced the following day.
Fox hunting with hounds is illegal in England, Scotland and Wales - but dogs can still be used to flush out foxes. There is no ban in Northern Ireland.
Mr Hunt wrote in the Daily Telegraph: "[Hunting] is part of the countryside. And we have to recognise that in terms of the balance of the countryside. You know, it's part of our heritage."
He added that he would support a free vote among MPs when there was a majority in Parliament "that would be likely" to lift the ban in England.
Fox hunting: What is the law?
- In Scotland, the hunting of foxes with hounds was banned under the Protection of Wild Mammals (Scotland) Act, which was passed in 2002
- In 2004, it was made a criminal offence in England and Wales to hunt down and kill a wild mammal with dogs
- In order to prosecute, a person must be shown to be intentionally hunting the animal
- Pursuit of live animals has been replaced by trail hunting, which sees hounds and riders follow a pre-laid scent along an agreed route
- An unlimited fine is currently the most severe punishment available
- Labour has promised to toughen up the ban on hunting with dogs in England and Wales, saying it will consult on jailing those caught breaking the law
- There is no ban on fox hunting in Northern Ireland
- There are 179 packs recognised by fox hunting's governing body, the Masters of Foxhounds Association (MFHA): 171 in England and Wales and eight in Scotland
When questioned on BBC Radio 4's Today programme, Mr Hunt appeared to play down his comments, saying he had been "restating the position in our manifesto from 2017".
He added he did not believe there would "ever" be a majority among MPs in favour of overturning the ban.
The foreign secretary avoiding saying whether he believed fox hunting was cruel, instead calling it "not my thing".
Shadow environment secretary Sue Hayman said: "It is incredible and shocking that Jeremy Hunt, who aspires to be the next prime minister, can not bring himself to even acknowledge the barbaric cruelty of fox hunting.
"Once again it shows this Tory government's lack of understanding and sincerity on issues of animal welfare."
Campaign group the League Against Cruel Sports said Mr Hunt was "out of touch with public opinion" in offering a vote.
Last year, Prime Minister Theresa May dropped a Conservative manifesto pledge to hold a vote on the fox hunting ban.
Her predecessor, David Cameron, had promised a free vote on whether to repeal the ban in England and Wales when he was in power, but it never materialised.
Mr Hunt and Mr Johnson, a former foreign secretary and London mayor, face a hustings of Tory members in York on Thursday evening.
Ahead of this, writing in the Sun newspaper, Mr Johnson argued that more police were needed to "end the current crime wave" across the UK.
He said 20,000 extra officers would help in "rounding up the leaders of the county lines drugs gangs" and help with "proper stop and search".
Mr Johnson added that it was "nonsense" to suggest stop-and-search powers were discriminatory and that "the most loving thing we can do is take the knives off the streets".
But Sir Tom Winsor, Chief Inspector of Constabulary, said although more officers were needed, some of the £1.1bn promised by would be better spent on new technology.
Between September 2009 and September 2017, police forces in England and Wales lost more than 20,000 officers - a drop of 15%. Numbers have remained more or less stable in Scotland since 2010.
Speaking on Today, Mr Hunt repeated an earlier promise to have an external investigation into alleged Islamophobia, saying it was important to "root it out" of the party.
On his plans for social care, he said while more government cash was required, people should be "encouraged and incentivised" to save for their own care.
Both candidates have been challenged by a House of Lords committee to bring forward plans to tackle the social care crisis if they become prime minister.
Mr Johnson is ahead of his rival in terms of donations. He has garnered £348,500 since the beginning of May, compared with Mr Hunt's £185,945, according to Parliament's register of interests.