Mr Johnson announced this week that, from September, people will have to provide proof of Covid-19 vaccination to enter venues such as nightclubs.
However, Clapton – like many others – feel the measures are discriminatory towards the individual and clubs.
In a statement released via architect, film producer and anti-vaxxer Robin Monotti Graziadei's Telegram account, Clapton said: 'Following the PM's announcement on Monday the 19th of July 2021, I feel honor bound to make an announcement of my own.
'I wish to say that I will not perform on any stage where there is a discriminated audience present.'
He added: 'Unless there is provision made for all people to attend, I reserve the right to cancel the show.'
Clapton, 76, has two scheduled shows at London's Royal Albert Hall in May 2022.
The Wonderful Tonight singer has been vocal throughout the Covid pandemic, releasing anti-vaccination and anti-lockdown statements and songs.
He collaborated with classic rocker and Van Morrison, who is against the jab, on the track Stand and Deliver in December.
Lyrics include: 'Do you wanna be a free man / Or do you wanna be a slave?' and 'Dick Turpin wore a mask too.'
In May, the musician detailed the 'disastrous' reaction he had to the vaccine and slammed what he branded 'propaganda' that 'overstated' the safety of the jab.
The Medical Healthcare Products and Regulatory Agency has previously reiterated that the benefits of the Oxford-AstraZeneca jab outweigh the risks for most people, while Mr Johnson and medical director of primary care at NHS England Dr Nikita Kanani also previously assured people the Oxford vaccine is safe.
However, Clapton spoke of the 'severe reactions' he had for 10 days following his first dose of AstraZeneca in February but noted he eventually recovered.
Metro.co.uk has reached out to Eric Clapton's rep for comment.
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