The "convincing" scam usually contains a link to a website – through text or email – that asks for personal details and payment to cover an admin fee.
Similar methods are used in the phone calls where people pretend to be from the NHS and asking for the money, despite Covid-19 passes being free.
Councillor Jasbir Jaspal, Wolverhampton Council’s cabinet member for public health and wellbeing, said: "Criminals are using Covid-19 as a way of making money out of unsuspecting people, conning them for payments that are unnecessary, so it is important that people are warned about this, and are alert to it.
"The NHS App and the Covid-19 Pass are completely free to use and you will never be asked to give out your financial details or make a payment to access them. If you are, it is a scam.
"So, if you receive one of these emails, texts or calls, the best thing to do is to report it straight away. Fake Covid-19 Pass certificates are also being sold online and on social media and these are not real either."
A Covid pass shows people's coronavirus vaccination details, which they may be asked to show if they travel abroad – or at events and venues in England asking for proof of the status.
It is available to anyone over the age of 16 once fully vaccinated. Passes are also available if a person has recorded a negative PCR test within the last 48 hours, or for up to six months after a positive PCR test. Passes can be requested online at nhs.uk/nhscovidpass by calling NHS 119.
Anyone who receives a suspicious Covid-19 pass email should report it by forwarding it on to firstname.lastname@example.org. Calls can also be reported through this email. Texts can be reported by forwarding to 7726, which is free of charge.