Andy Burnham has said he would like to shut down Tesco supermarket branches in which fail to enforce face mask use by customers.
The Greater Manchester mayor revealed he has asked for new powers from government to close venues which do not observe rules designed to make them Covid-secure.
And he named the grocery giant as one chain which he believes has failed to act responsibly.
Under government rules, shoppers in England - with certain exemptions - must wear face-coverings while inside a store or supermarket, with police able to impose a £200 fine on those who do not.
The official guidance does not require stores to enforce the rule, but says they should “take reasonable steps to promote compliance with the law”.
But Mr Burnham told the House of Lords Economic Affairs Committee he believed the companies should be responsible for making sure shoppers follow the rules.
Watch: What could a road map out of coronavirus restrictions involve?
And he said that targeting venues which fail to make themselves Covid-safe could have more effect on beating the virus than the blanket closure of pubs bars and other businesses imposed by the government when Manchester went into tier 3 restrictions last week.
Giving evidence to peers on his negotiations with ministers, Mr Burnham said: “We argued for a more targeted approach to venues.
“We asked for summary closure powers of all business premises that were not deemed to be Covid-safe.
“I'm not just talking about pubs and restaurants. There are large supermarkets in Greater Manchester who in my view have not properly implemented the requirements around face coverings.
“And I'll name one chain - Tesco. I don't believe they have taken the responsible attitude to this issue.
“They've said it's not for them to enforce. I think it is for those organisations to enforce.
“So we have asked for targeted summary closure powers to be able to temporarily close venues which we do not believe are adhering properly to the protocol.
"And we think that would be a better approach than just blanket restrictions that may or may not have the impact that the government wants on the virus.”
Mr Burnham said that Manchester’s experience over the past few months suggested that targeting venues which refuse to enforce face mask use, coupled with shielding of the most vulnerable residents, could work better than the restrictions on social and economic life imposed by the government on the city
“We’ve been under those restrictions for three months,” said Mr Burnham. “So together with Leicester, we probably have more experience with local restrictions than anywhere.
“And our view is they may have dampened the number of cases, but they haven't actually succeeded in turning the situation around.
“We think the ability to act in a more targeted way, both in terms of shielding but also in terms of enforcement, would be a more sensible approach.”
Watch: Boris Johnson says he emphasises with Andy Burham as Greater Manchester moves into Tier 3