Boris Johnson wants the world population to be inoculated against Covid-19 by the end of next year and is encouraging wealthy nations to adopt vaccine passports.
The UK prime minister is also aiming to open talks on a world treaty that details how nations should prepare for future pandemics, ahead of Britain hosting the G7 summit in Cornwall next month.
"We need to have agreements on issues such as vaccine passports, Covid status certification and the rest," Mr Johnson told Canadian broadcaster CBC.
"There has to be some sort of agreement then, at the G7 level to start, on how travel and passports are going to work."
Mr Johnson said the world must set a more ambitious target to inoculate the global population, adding "we need to get this done by the end of next year”, rather than in 2024 or 2025.
He suggested a new treaty on pandemic preparedness could help prevent what he said was the “balkanisation” of global co-operation.
During the Covid-19 pandemic, many nations were slow to share stocks of protective equipment, medicines and vaccines.
"We've got to do better than this. Vaccination has got to be a global enterprise," Mr Johnson said.
The prime minister’s comments come as scientists say a third wave of infection may have already started in the UK.
On Friday, the UK recorded more than 4,000 daily Covid-19 cases for the first time since April 1, with the majority of new cases now the more infectious variant first identified in India.
The R number – which measures how many people an infected person will pass the virus on to, on average – is fully above 1 for the first time in months.
Prof Ravi Gupta from the University of Cambridge, a government adviser, said the outlook was not good.
“If things go as I think they are going to go, we will likely end up with a third wave. It will be a big wave of infections and there will be deaths and severe illness,” he said.
“It will put pressure on the [National Health Service] at a time when we are trying to get back to normal and it is going to require a redoubling of efforts from the government to step up vaccination and to look at boosting of waning [immune] responses.”
Meanwhile, Prof Martin McKee from the London School of Hygiene and Tropical Medicine said he believed the third wave was already under way.
“We can already see that the current measures are not stopping cases rising rapidly in many parts of the country. This looks very much as if we are now early in a third wave,” he said.
“Unless there is a miracle, opening up further in June is a huge risk. The rise in cases we are seeing now should cause a reassessment of the most recent relaxation.”
Both scientists suggested Mr Johnson should delay the final stage of lockdown easing on June 21, when most social distancing restrictions are due to end.
Environment Secretary George Eustice said Mr Johnson would make a decision on the next step in the coming weeks.
“The prime minister said all along he is going to take this one step at a time,” he told the BBC.
“The rates are going up again, slightly but from a low base, and that is probably to be expected given there is still a significant number of younger people who are out and mixing that haven’t had the vaccine.”