He said his team had believed his chance to be “a very long shot” at the start, but that had changed.
The foreign secretary said he believed “so many people” had switched to support him after hearing him speak.
Meanwhile, Mr Johnson has been pledging to support the “wealth creating sector” in the UK.
On a visit to a sausage factory in North Yorkshire, he said companies in the UK showed “incredible diversity and resilience”.
He added that boosting business was the “way to pay for great public infrastructure, social care, all the things we need”.
Tory members have begun to vote for their choice, after receiving their polling papers earlier than expected.
The winner of the contest will be announced on 23 July and will take over from Theresa May on 24 July.
Talking to BBC political editor Laura Kuenssberg, Mr Hunt said: “I think at the start of the campaign the people around me thought this was a very long shot. Now it feels like much more of a contest.
“We’ve had some surprises. We had the surprise of Trump. We had the surprise of Brexit. I think we could have the surprise of the Conservative Party leadership election too. I think there’s all to play for.”
He said he believed some Tory members had switched support from frontrunner Mr Johnson.
“It’s just a question of whether we’ve done enough in the short time we’ve had, but I think it’s completely doable,” he said.
Mr Hunt also said he “would like longer” to campaign for the leadership, adding that he was “disappointed that Boris hasn’t wanted to do head-to-head TV debates until after most people would have voted”.
“I think members would have liked to have seen us both in action,” he said. “But that’s his choice.”
The candidates are set to face each other in an ITV debate on 9 July and at an event hosted by the Sun newspaper and talkRADIO on 15 July.
They will also be interviewed by Andrew Neil on BBC One on 12 July.
Mr Hunt has offered MPs a free vote on lifting the ban on fox hunting in England and Wales if he becomes prime minister.
He said he was “just being honest” about how he had voted on the issue in the past, “but that’s not what I’m going to change as prime minister”.
He said his priorities included rural broadband and protecting farmers in the event of a no-deal Brexit.
On Brexit, Mr Hunt repeated that he would leave the EU without a deal “if there isn’t a deal to be done” and that decision would be made by the end of September.
Later, at a leadership hustings in York, Mr Hunt said councils needed more money for social care and there needed to be incentives for people to save for their futures, “just as we save for a pension”.
He said the government should also consider incentives to encourage people to look after elderly relatives in the family home.
“There are 420,000 households in our country that are three-generation households where granny, mum and dad and the kids all live under the same roof. I think that is a good thing,” he said.
“I am not saying we all want to live with our mother-in-law. But I think that three-generation families are a wonderful thing.”
When asked about a committee report that called for an immediate £8bn cash increase and a move to a free, NHS-based system, for social care, Mr Johnson said the report was “thought-provoking”.
“There’s no question that they’re putting their finger on an issue that we have to address,” he told the hustings.
“It simply cannot be fair that Alzheimer’s or dementia are not properly funded in the way that other illnesses at the end of life are under the NHS.”
Pressed on whether he would commit to a green paper on the topic, he said: “I will certainly commit.”
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