Criticising Sir Keir Starmer’s policy, Sir Lindsay said “competition” from a new, fully-elected body could threaten the supremacy of the House of Commons.
“I don’t think we should have a second house that’s elected. I don’t want competition,” the former Labour MP told LBC.
“Who’s got supremacy? At the moment, it’s very clear the House of Commons, the elected House, has supremacy, once you have a second house that’s elected, then you’re into an arm wrestle about who has the power.”
The Speaker suggested Labour revise its plan to abolish the unelected chamber and opt for less radical reform.
“What I would say is, if people want to reform it, please do – but do not have a second elected house,” he said. “We don’t need the competition. Supremacy is going to remain with the Commons. That is what people recognise as the senior house.”
Sir Lindsay added: “We don’t want any arm struggles. We want to very clear, distinct powers separated very, very easily straightforward, but not via an election.”
Labour would aim to abolish the unelected Lords “as quickly as possible” – ideally within its first term, Sir Keir said earlier this month.
But the party’s leader did not commit to a timeframe for a new democratic assembly of nations and regions to replace the Lords, stressing discussions are pending on when “exactly” it would come to pass.
The proposal forms part of Labour’s blueprint for a New Britain, outlined in the report of its commission on the UK’s future – headed by ex-premier Gordon Brown.
Mr Brown’s panel called for a new constitutional law setting out how political power should be shared, with a requirement for decisions to be taken “as close as meaningfully possible” to the people affected by them.
But the former Labour prime minister has insisted the current upper chamber is now “indefensible” and has to go.
He warned that the issue could “come to a head” when Boris Johnson publishes his resignation honours list, which is expected to include a number of controversial new peers.
Mr Johnson’s former chief of staff Dan Rosenfield faces claims that he bullied junior members of staff during his time at Downing Street.
Some No 10 employees who worked with Mr Rosenfield are attempting to stop him receiving a peerage after being nominated by Mr Johnson, according to The Times – claiming it would be “entirely inappropriate” because of his conduct.
But some staff said the claims of bullying were “laughable”. One told the newspaper: “This is the latest weak attempt from a group of disgruntled former political aides … It is so obviously political as to be laughable.”