Experts believe restrictions could be tightened in a number of ways including making it a legal requirement for supermarkets to require face masks.
Other measures could include closing nursery schools, shutting places of worship, banning all household mixing and making face masks compulsory in offices, queues and other crowded places outdoors.
Mr Johnson said: “We need to enforce the rules in supermarkets.
“When people are getting takeaway drinks, in cafes, then they need to avoid spreading the disease there, avoid mingling too much.”
Vaccine minister Nadhim Zahawi echoed this sentiment on Monday when he said: “We are concerned that, for example, in supermarkets we need to make sure people actually wear masks and follow the one-way rule in supermarkets.
“And, of course, when they are at capacity – to operate safely, people wait outside supermarkets.
“We don’t want to go any tougher, because this is a pretty tough lockdown.”