Andrew Jones MP said he would take forward the idea proposed by the DUP's Jim Shannon, which would ban drivers with multiple offences from hiring cars with anything bigger than a two-litre engine.
Speaking during transport questions in the Commons, Mr Shannon said: "Would the secretary of state agree that anyone with points on their licence that would indicate a number of offences should be excluded from rental cars above a two-litre engine?
"Would he consider co-operation with the policing and insurance boards on this issue?"
In reply, Mr Jones said: "He makes a very interesting point. I don't think we can necessarily exclude people from a marketplace, but of course all the rental companies do have access to a driver's records.
"I will take that idea forward."
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New rules came into effect last year changing the documents needed when hiring a car.
The Driver and Vehicle Licensing Agency (DVLA) scrapped the paper counterpart to the driving license photo card.
The document contained a history of drivers’ convictions and penalties.
Instead, drivers now have to generate a code using the DVLA website to demonstrate their background to car hire firms.
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Mr Jones has previously praised the car rental industry for playing an “important role” in the transport industry.
His comments were in response to a survey by British Vehicle Rental and Leasing Association earlier this year that suggests car hire customers have more sustainable travel habits than the average licence holder.
He said: “Car hire gives people extra choice for getting around and it can help make travelling healthier, more sustainable and accessible to all. We want to help to reduce congestion and emissions and these figures show that car rental customers help deliver that ambition as they tend to walk, cycle and use public transport more often.”