Health Minister Lord Howe said changes to the NHS charging regime for overseas visitors and migrants "is the first step".
Ministers have said no one will be turned away from an accident and emergency department, but there will be a bill to pay afterwards for those from overseas.
Lord Howe said:
Having a universal health service free at the point of use rightly makes us the envy of the world, but we must make sure the system is fair to the hard-working British taxpayers who fund it.
We know that we need to make changes across the NHS to better identify and charge visitors and migrants. Introducing charging at primary care is the first step to achieving this.
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Harry will also pay tribute to a British soldier killed by an elephant during anti-poaching operations in Malawi.
The family are making an official visit to South Africa.
The duke, who has admitted “I feel more myself in Africa”, is about to begin a tour of South Africa with the duchess and Archie.