The vaccine is being researched and developed by Moderna. The new vaccination will be administered via a single inoculation. It has been reported that it will be ready by next winter.
The development will mean people will not have to get two separate vaccines to offer protection against Covid and flu.
Cases of people developing both Covid and flu at the same time are rising, particularly during the winter months.
Speaking to the Daily Mail, Moderna UK chief Darius Hughes said: “Our No 1 priority for 2022, after getting the right Covid vaccine for the Omicron variant, is to try and really drive forward our flu and RSV programmes.
“This is to see if we can get a combination, single-dose respiratory vaccine.
“It would be just fantastic for patients if you could have one appointment go along and have that jab done.
“Then you’re protected against all of those major respiratory diseases for the winter.
“The benefit for the NHS, and all the vaccination services, and ultimately patients, we think, is going to be huge.”
However, fears of a combination of flu and Covid have been played down by most scientists.
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Doctor Simon Clarke, a microbiologist at Reading University, speaking to the Daily Mail said there were “fairly low” flu numbers in the UK so the risk of coinfection along with coronavirus is “not currently high”.
He said: “Contracting both viruses simultaneously increases the risk of being seriously ill.
“But risks can be mitigated by getting vaccinated and boosted against both diseases.
“Allowing increased numbers of either infection increases the risk of contracting both of them.
“With testing for both viruses uncoupled as they are, the numbers of confirmed flu infections is probably a substantial underestimate and it is possible that the risk of dual infection is much greater than we realise.
“I would be surprised if the current number of flu and Covid co-infections was below a thousand.”
When CBS tweeted that Los Angeles had recorded its “first case of flurona” this week, the broadcaster was criticised by Dr Angela Rasmussen, a virologist at the University of Saskatchewan.
She tweeted: “Stop this ‘flurona’ BS. Co-infections of circulating respiratory viruses happen all the time. Co-infections of influenza and SARS-CoV-2 were reported in early 2020. This is almost certainly not the first time this has occurred in LA or anywhere.”