Jeremy Hunt is pledging to set out his plans to overhaul the social care system within 100 days of entering No 10 if he wins the Tory leadership race.
Mr Hunt, a former health secretary, said he regarded reform of the adult social care system as “unfinished business”.
He said he would publish his proposals – including incentives to encourage people to save for their care needs in the way they do for pensions – in a green paper.
His promise came after a House of Lords committee warned it would require £8 billion a year to fix the underfunding in the system.
Meanwhile his rival, Boris Johnson, has said he will make Britain “the greatest place on Earth” if he succeeded in gaining the keys to 10 Downing Street.
With some party members already reportedly having received their postal voting ballot papers – two days earlier than previously expected – Mr Johnson said he was “fighting for every vote”.
“I’m taking nothing for granted,” the former foreign secretary – seen as the clear front runner – told The Daily Telegraph.
“It’s time to drive for the line, keeping the ball at the back of the scrum. I’m fighting for every vote and campaigning flat out and will be from now until the end.”
Mr Hunt, the current Foreign Secretary, said that under his plans the government would guarantee no-one would have to sell their home to cover their care costs.
In return for people being incentivised to save for their needs, he said the state would pledge to cap the cost of care.
Tax breaks for home adaptations and extensions would be introduced to aid families looking after elderly relatives, and more money would be made available to bring health and social care together under the initiative.
The initiative would see a 10-year funding plan for the social care system, including additional state funding for local councils from the £26 billion “headroom” reserves Chancellor Philip Hammond has set aside.
He said: “That’s why guaranteeing older people dignity and respect in their final years is unfinished business for me.
“If Conservatives stand for anything it has to be dignity and security for our elderly generation as their lives draw to a close.
“I want to bring health and social care together, and back those unsung heroes caring for their own family members for free.”
Campaigning on Thursday in Yorkshire, Mr Johnson also acknowledged the issue had to be addressed but has yet to say how he would do so.
Meanwhile the pro-EU Justice Secretary David Gauke has warned that Parliament would find a way to prevent Britain leaving the EU in a no-deal Brexit.
Both contenders have said they are prepared to walk away without a deal if they cannot secure a new agreement with Brussels on the terms of Britain’s withdrawal.
However Mr Gauke told Parliament’s The House magazine: “If I was to speculate on it, given that we have an activist Speaker, given that there is a parliamentary majority against no deal, a way will be found.”
He was backed by former minister Sam Gyimah who said some Tory MPs would even vote for a no confidence motion to bring down the Government to prevent no deal.
“It is not an option I am considering but I know there are other colleagues who will consider that, who will go that far,” he told BBC2’s Newsnight.
“There will be more than enough of us who will use every legislative means possible to stop no-deal.”
Appearing at a leadership hustings in York, Mr Johnson made clear he would not brook any dissent from ministers on his commitment to leave the EU by October 31, with or without an agreement.
“What I wouldn’t accept is people actively dissenting in public from that policy and undermining it,” he said.
His comments were seen as a thinly veiled swipe at Cabinet ministers like Mr Gauke, Amber Rudd and Greg Clark who infuriated Brexiteers when they suggested earlier this year they would be prepared to prevent no deal.