Michael Gove will allow EU nationals living in the UK at the time of the referendum to apply free of charge for citizenship if he becomes PM.
The Brexit-supporting environment secretary, who is running to replace Theresa May, will make an "open and generous" offer, sources said.
A Tory rival, International Development Secretary Rory Stewart, promised a "listening exercise" on Brexit.
So far, 10 Conservative MPs have said they will contest the party leadership.
The official race gets under way in early June, after Theresa May stands down - but jostling between candidates has begun. The winner, expected to be named by late July, will also become prime minister.
Sources close to Mr Gove, one of the leaders of the Leave campaign during the 2016 Brexit referendum, have told the BBC he is ready to accept proposals put forward by the Conservative MP Alberto Costa, who quit his government post over ministers' attitude to EU nationals living in the UK.
If chosen as the next Tory leader, it is said he would remove the requirement of EU citizens to provide proof of their right to be in the UK, getting rid of the "settled status" scheme.
Those living in the country would require documentation only for specific purposes, rather than being required to register.
Mr Costa welcomed Mr Gove's proposal, calling it "the morally right thing to do"
A source close to the environment secretary said: "This is simply the right thing to do - honouring the promise of Vote Leave that EU nationals studying, working and living in the UK were welcome to stay."
The leadership contest comes after Mrs May tried and failed three times to get her Brexit withdrawal agreement through the House of Commons.
She announced her resignation last week following an outcry within her party when she proposed a fourth vote by MPs.
The Conservatives suffered heavy losses in Thursday's European elections.
'Getting deal done'
Mr Stewart said: "I would like thousands of conversations up and down the country, co-ordinated on social media with all the results being brought together digitally.
"And then we come back into Parliament and we move very quickly to ban conversation about no deal, ban conversation about second referendum and focus on getting a deal done."
Meanwhile, housing minister Kit Malthouse has become the latest Conservative MP to join the race to become party leader.
Writing in the Sun newspaper on Tuesday, Mr Malthouse said the campaign "cannot be about the same old faces" and described himself as "the new face, with fresh new ideas".
The other declared candidates to replace Mrs May are:
- Home Secretary Sajid Javid
- Health Secretary Matt Hancock
- Foreign Secretary Jeremy Hunt
- Former Foreign Secretary Boris Johnson
- Former Leader of the House Andrea Leadsom
- Former Work and Pensions Secretary Esther McVey
- Former Brexit Secretary Dominic Raab