Around 40pc of those who moved in the past year ended up in contracts that were an average £1,200 a year above the advertised rate, according to think tank the New Economics Foundation.
Scotland, which has rent controls, saw the biggest percentage increase in rent in the same period, with costs rising 9.3pc.
The National Residential Landlords Association, an industry body, agreed with the proposals.
Policy director, Chris Norris, said: “Reports that some letting agents are encouraging payments above this are concerning. Whilst we agree with the principle of ending ‘bidding wars’ to tackle the actions of those who do promote such practices, more detail is needed about how this would work and be enforced.”
But others accused Labour of misunderstanding the rental crisis.
Chief executive of trade body Propertymark, Nathan Emerson, said: “Because you have such pressure on stock it [a bidding war] does not come from landlords, it comes from tenants offering above the price or saying they can pay a year up front.
“Agents are legally obliged to pass on all offers to the landlord. The property can only go to one person, those who lose out feel frustrated and bid higher for the next property and it goes on.”
“There is a longer term picture here so let’s see the details but I don’t believe it will solve the problem and my experience is landlords aren’t out there to get every penny they can, they just want consistency and security.”
Partner at law firm Gowling WLG Senay Nihat warned a ban on bids could push up listing prices instead.
“Clearly for renters this is a positive step, especially with an overheated market where rents are consistently rising, particularly in the midst of a cost of living crisis.
“For landlords of course this is an intervention in the market. If a tenant is willing to pay more, arguably they should be allowed to do so. You might therefore see landlords starting listings with higher rents to begin with.”
The proposals come in addition to rental reforms detailed in Labour’s manifesto.
The party has said it plans to reinstate costly energy efficiency targets and abolish no-fault evictions “immediately”.
Labour said it would give landlords until 2030 to comply with new “minimum energy efficiency standards” – a plan previously shelved by the Tories.
The targets, which included measures such as installing eco boilers and solar panels, would have cost homeowners thousands of pounds each.
The Labour party declined to comment.
