Under the scheme, drawn up by cross-bench peers and endorsed by Lord Cranborne and Tony Blair, 91 hereditary peers will retain their voting rights until the Government has introduced wider reform of the second chamber.
While Lord Strathclyde, the newly appointed Tory leader of the Lords following Lord Cranborne's sacking, made clear there was no official agreement between the Government and the Opposition, he secured a deal from William Hague that Tory peers would not be asked to vote against the concession on hereditaries.
However, he urged the Prime Minister to include the plan in the forthcoming Bill on House of Lords reform so MPs would have sufficient time to debate the proposal.
Presently the concession is likely to be introduced by cross-benchers as an amendment once the Bill has cleared in the Commons and goes to the Lords for debate.
Lord Strathclyde, a popular figure in Westminster, admitted he was facing a "great challenge" to lift the morale of Conservative peers, saying they were feeling "very unhappy and very bruised".
A new frontbench team will be appointed next week after a series of resignations in protest at Lord Cranborne's "brutal" sacking by Mr Hague.
Lord Strathclyde added he would seek to convince his party to go back to their role as an "effective and thorough" Opposition as soon as possible.
"It is going to be very difficult after what happened this week but we have an important role as a revising chamber and we should go back to that and put this all behind us," he told The Independent.
The Tories will continue to oppose the "closed-lists" system to be introduced under the European Parliamentary Elections Bill, which was reintroduced in the Commons earlier this week, after being defeated five times by defiant peers in the last parliamentary session.
Other key Government Bills, such as legislation on trade union recognition and welfare reform, are also likely to come under fierce scrutiny by peers.
"I have never used the term zero tolerance and there will not be opposition for the sake of it but we will put down amendments where we disagree."
Lord Strathclyde, who accepted his new post only under the condition that he would not criticise Lord Cranborne in public or in private, stressed his continuing friendship with the sacked former Leader of the Lords. "Robert [Cranborne] will remain a great friend of mine," he said.
Previously the chief whip in the Lords, Thomas Galloway Dunlop du Roy de Blicquy Galbraith is the second Lord Strathclyde, inheriting in 1985, at the age of only 25, the title awarded to his grandfather 30 years earlier.
A former Lloyd's insurance broker, he quickly made his mark in the Lords, first in the Whips' Office, then as a minister at the Department of Trade and Industry in 1994. He was sent back to the Whips' Office as Government chief whip, where he served during the last three years of John Major's government.Reuse content