The video, shot at London’s Chelsea and Westminster Hospital, aims to “show others that getting the life-saving jab is easy and safe”, according to a statement.
The clip shows the pair appearing to audition for a part in an advert promoting the jab.
In the video Sir Elton, 73, said: “The more people in society that get vaccinated, the more chance there is of eradicating the national Covid pandemic.
“It is really important to know that the vaccines have all been through and met the necessary safety and quality standards.”
After apparently being given the injection, Sir Elton then references his 1983 hit I’m Still Standing by saying: “As you can see, I’m still standing, yeah, yeah, yeah.”
The director then responded by saying: “Thanks Elton, we will let you know.”
Sir Elton replied: “Well at this short notice you won’t find anyone bigger.”
The video then cuts to Sir Michael, 87, as he appears to be given the vaccine.
The actor said it “didn’t hurt” before saying: “Not many people know that.”
Afterwards, the director can be heard telling someone to “let the little fella know he didn’t get the job”.
The video has been released after the NHS asked anyone aged 70 and above who has not yet been vaccinated to come forward to receive one.
The campaign’s co-ordinator Professor Lord Ara Darzi, who is co-director of the Institute of Global Health Innovation at Imperial College London, said Sir Elton and Sir Michael are “trusted faces” who are “loved” by the public.
“I asked them [to take part] and they were extremely generous in giving us the time to do this, which I was very grateful for, and I hope it does the job in convincing the population that vaccines for Covid is the best way out of this pandemic,” he told the PA news agency.
“They were extremely keen to help in any way possible.”
Lord Darzi added he wanted the video to be “humorous and engaging” and he hopes people who are sceptical about the vaccine will change their minds.
“I think many people who have had two thoughts about this will think again,” he said.
NHS chief executive Sir Simon Stevens said: “Well over nine out of 10 people aged 75 and over have already taken up the offer of their NHS Covid vaccination, and now we want to encourage everyone in the high priority groups to do so.
“That’s why we’re grateful to Sir Elton and Sir Michael for helping the NHS promote vaccine uptake, building on the fast and efficient programme to date, which has seen the NHS already vaccinate one in four adults across England.”