Lynda Hamlyn, chief executive of NHS Blood and Transplant, said more than 10,000 people currently need a transplant but due to a lack of organs about 1,000 people die every year before they can have one.
''In a situation where there are not enough organs to treat the citizens of the very country donating them, the priority must be to ensure a fair and open system of allocation and treatment and the necessary level of public reassurance that this is the case.''
Elisabeth Buggins, former chairwoman of the Organ Donation Taskforce, carried out an inquiry after allegations in a number of newspapers that organs from NHS donors were being given to patients from countries such as Greece and Italy.
It emerged that more than 700 transplants, mostly liver transplants, had been carried out on non-UK patients over the past decade.
In total, 631 of those transplants used organs from dead donors and, of those, 314 were from outside the EU.
It is not clear how many of those paid privately.
The inquiry found no evidence of wrongdoing in how organs were allocated to these patients, but concluded that in the interests of fairness no one should be able to pay for such operations.
It also says that rules should be tightened on which EU citizens are entitled to transplants on the NHS.
Under EU law, some patients can receive treatment in other countries, if approved by their healthcare system, which then foots the bill.
But the NHS needs to be more cautious when checking eligibility under these rules and it is likely that there are patients currently receiving treatment who should be refused, Mrs Buggins said.
She also recommended that the NHS works with other countries in the EU to develop their own transplant programmes.
And any reciprocal arrangements with transplant networks in other countries need to be reviewed, she advised.
Surgeons will still be able to carry out private work using organs from living donors, for example with kidney transplants and some liver transplants.
Last year 3,500 transplants were carried out in the UK.
The Department of Health, which covers England, accepted the recommendations and said immediate steps would be taken to ban all private clinical practice involving solid organs donated after death.
Agreement for this still needs to be reached with the administrations in Scotland, Wales and Northern Ireland but is expected to come into force across the UK in October.
Elisabeth Buggins said the report aimed to make more organs available for UK residents.
''Confidence in the transplant system should increase once money is removed from the equation, decisions are transparent and accountability clear; confidence we know is necessary if the number of organ donors is to rise to match the best in Europe.
Health Minister Ann Keen said they would implement the recommendations to ensure a UK system that is ''fair and transparent''.
She added: ''The report highlights the complexity of European law in this area and we will take immediate action to provide guidance for the transplant community and reassure the public of the integrity of our transplant programme.''
The British Transplantation Society welcomed the report, saying it would provide ''further reassurance'' that priority for a transplant was given to those in greatest need.
It added: ''We welcome the proposal to clarify the arrangements for the treatment of non-UK patients and would encourage the Health Secretary to pursue this without delay.''
Joyce Robins, co-director of Patient Concern, also hailed the proposal, arguing that any suggestion of private payment seriously undermined the entire transplant programme.
''Why should we sign up as organ donors if our organs can then be sold to the highest bidder? The law rightly prevents us from selling our own organs, so it is an outrage that hospitals can boost their income by doing so, while UK residents die for lack of organs.''