Health Secretary Jeremy Hunt is understood to be pushing for the measures and Government insiders claim he is winning over allies in Downing Street.
But Prime Minister Theresa May is understood to be unconvinced.
Tory MPs have warned that the proposals risk provoking accusations of meddling and could increase the cost of living for the country’s hard-pressed households.
But MPs backing the move argued that “nanny state” objections should not apply to policies designed to protect children.
Sarah Wollaston, Tory chairwoman of the Commons Health Committee, said: “Adults have a right to make decisions about their health but when it comes to children, particularly with these profound inequalities, it’s absolutely right for the Government to make progress and pull the levers that will make a difference.”
Tory MP Philip Davies, a former marketing manager at Asda, said: “This is a ridiculous proposal.
“Why would the Government want to increase the cost of living for people?
“I worked for Asda for 12 years before becoming an MP and so I saw that customers love two-for-one offers.
“If somebody wants to buy something sweet, why shouldn’t they be allowed to do that?
“The Government has plenty of more important stuff to deal with before poking its nose into every nook and cranny of people’s lives.
“This is the sort of nonsense you might expect from a socialist government, not a Conservative one.”
Celebrity endorsements of unhealthy food products would also be prohibited under the proposals.
Mrs May’s spokesman yesterday dismissed reports of the crackdown as “speculation”.
He added: “The Government has an obesity strategy, which has already been set out.”