This could include accidents such as the Paddington rail disaster, which left 30 dead and hundreds injured, if somebody could be held liable.
But the organisation which represents NHS managers said that it might take too much time and money to force insurers to pay out.
Department of Health figures put the cost of treating the average car accident victim in hospital at nearly £3,000.
At present, people who have been injured through someone else's negligence can take legal action to have their medical and nursing bills paid by those responsible or their insurers.
However, if the hospital care is provided free by the NHS, health bosses are entitled to try to recover the cost only when the injuries were caused in a traffic accident.
And at present, the NHS does not generally try to recover the money involved.
The Law Commission is recommending that the health service should be given more powers to recover treatment costs in future.
This could bring in about £100m a year for the government.
But it would also mean that motorists and employers would face higher insurance premiums.
Wider range of dependents
In fatal accident cases, the person who caused the death may have to pay compensation to the dead person's financial dependents.
The Law Commission recommends that those dependents could now include partners of the same sex, as well as cohabitees, engaged partners and siblings.
Even godchildren, friends or anyone else able to show that they were financially dependent on the dead person should also be entitled to claim damages, recommends the report.
Now the Law Commission has published its findings, it is the Government's decision as to whether they should be made law.
However, Stephen Thornton, the chief executive of the NHS Confederation, which represents health service managers, said that while it was logical for the NHS to have the right to claim money, the costs of collection could almost outweigh the revenue.
He said: "What we would not want to see is the Treasury assuming this as an annual source of funds and docking that from our allocations. Any fund being recouped needs to be over and above mainstream funding, as in the way lottery funding is allocated to deserving causes."