Calling such an idea "discriminatory", Mr Zahawi suggested Brits could instead use letters from GPs if they needed to meet the travel requirements of other countries.
Speaking on The Andrew Marr Show, Mr Zahawi said: "There are several reasons why we're not doing that, one because vaccines are not mandated in this country as Boris Johnson has quite rightly reminded parliament - that's not how we do things in the UK, we do them by consent.
"We yet don't know what the impact of vaccines on transmission is and it would be discriminatory. Of course, you have the evidence that you've been vaccinated held by your GP and if other countries require you to show proof of that evidence then that is obviously up to those countries but we have given the first dose to 11.5million people and we have no plan of introducing a vaccine passport."
Speaking to Sky News, the vaccine minister added: "If other countries obviously require some form of proof, then you can ask you GP because your GP will hold your records and that will then be able to be used as your proof you've had the vaccine. But we are not planning to have a passport in the UK."
The idea of introducing Covid vaccine passports is under much debate. For example, tourist officials in Greece have previously suggested the country could be open to tourism from May, providing visitors can prove they've had the jab.
In the UK meanwhile, shadow business secretary, Ed Miliband, said vaccine passports "may be necessary".
"We should be open to this but there are complications to do this vaccine passport. Is it just for international travel? Is it for as you go about your business in your society?" he said.
"We are five or six weeks on from the South African variant being discovered. We still don't have a quarantine system in place. We still don't have a plan for a comprehensive quarantine system, so let's proceed a step at a time, and of course, look at the vaccine passport - but let's get the quarantine right and let's get the rollout right too."
The news comes as the government says it is confident all over 50s will have been offered the vaccine by May.
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