In a report on reforming the personal injury claims process, the insurance firm said claims management companies and lawyers add significant costs in fees.
A "typical" whiplash claim currently costs around £2,500 in legal fees, but Aviva said handling claims directly would cut around 50% off the price.
There is no difference in the compensation awarded to the injured party if handled directly or through third parties, the company claimed, but it could lead to a fall in premiums of around £60 a year for the average driver.
The cost of buying motor insurance has risen by 80% since October 2008, and some 550,000 whiplash claims are received by insurers every year.
According to the AA, the average motor premium in the UK is £789, and Aviva said 15%, or £118, of this goes towards paying for whiplash claims.
Aviva's claims director, Dominic Clayden, said that removing "interested parties" would make the system more efficient.
"Our figures ... show that dealing direct with an insurer directly results at least as much compensation for the claimant and has the advantage of being quicker - meaning their treatment and rehabilitation can start almost immediately," he said.
"It would also prompt a significant reduction in the costs of the current system which would benefit all UK motorists, who will begin to see a reduction in their premiums."
But the president of the Law Society questioned whether insurers would be fair to claimants.
"Who do you trust to give you a fair deal? A lawyer working for you, or an insurer working for the person or company in the wrong whose main interest is minimising what they pay you?" Lucy Scott-Moncrieff said.
She added: "You need someone independent to make sure that you get what the law entitles you to have.
"It appears that Aviva want to strip us all of our rights so they can make more profits."
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