Health Secretary Sajid Javid says he has already put plans in place to give the most vulnerable groups a third or booster Covid jab early next month.
But the head of the Oxford Vaccine Group said doses needed to "go where they can have the greatest impact" - to protect unvaccinated people abroad.
The UK itself currently had no reason to panic, Prof Sir Andrew Pollard said.
Mr Javid said he would wait for final advice from the Joint Committee on Vaccination and Immunisation (JCVI).
"My plan is to offer the flu vaccine, especially to over-50s, at the same time," he said.
A bigger flu season than normal is expected, with scientists warning respiratory viruses will make a comeback this year after disappearing last year during lockdown.
NHS bosses had previously said they needed lots of warning of a Covid booster programme in order to plan the logistics of vaccinating millions of people against both viruses.
However, Prof Pollard, who led the team that created the Oxford-AstraZeneca vaccine, said decisions on whether to give boosters "should be scientifically driven".
"The time we would need to boost is if we see evidence that there was an increase in hospitalisation - or the next stage after that, which would be people dying - amongst those who are vaccinated," he said.
"And that is not something we are seeing at the moment."