Professor Sir Andrew Pollard, director of the Oxford Vaccine Group, referred to the idea as “mythical” and warned that a vaccine programme should not be built around the idea of achieving it.
It comes as three-quarters of adults have now received both doses of a Covid-19 vaccine.
The Department of Health and Social Care (DHSC) said that a total of 86,780,455 doses had been administered in the UK.
This includes 47,091,889 people who have receiving a first dose (89 per cent) and 39,688,566 people who have received both doses (75 per cent), according to DHSC statistics.
Herd immunity is when enough people become resistant to a disease – through vaccination or previous exposure – that it can no longer significantly spread among the rest of the population.
Sir Andrew told a session of the All-Party Parliamentary Group on Coronavirus (APPG): “We know very clearly with coronavirus that this current variant, the Delta variant, will still infect people who have been vaccinated and that does mean that anyone who’s still unvaccinated, at some point, will meet the virus.”
Sir Andrew said while vaccines might “slow the process” of transmission down, they currently cannot stop the spread completely.
He said: “I think we are in a situation here with this current variant where herd immunity is not a possibility because it still infects vaccinated individuals.”
The Oxford Vaccine Group director predicted the next thing may be “a variant which is perhaps even better at transmitting in vaccinated populations”.
He said: “So, that’s even more of a reason not to be making a vaccine programme around herd immunity.”
The parliamentary group’s chairwoman, Liberal Democrat MP Layla Moran, called on the government to “re-evaluate their approach” and put together a “new and comprehensive” plan to mitigate the risks from new variants.
She said: “The worrying evidence we heard today suggests that given the emergence of new variants, including vaccine-resistant ones, achieving herd immunity is just a pipe dream.
“The government’s plan to learn to live with Covid cannot become a byword for abdicating responsibility to the most vulnerable.
“Ministers must re-evaluate their approach and come up with a new and comprehensive, long-term plan to mitigate the risks posed by Covid and new variants.
“That must include showing moral leadership and stepping up efforts to vaccinate the world.”
Sir Andrew said while there might be “bumpiness” in the next six months, there is also likely to be “increasing confidence” with the UK’s situation with the virus.
He told the APPG: “I think this next six months is a really important consolidation phase and in that shift from the epidemic to the endemic, which is the ‘living with Covid’.
“That doesn’t mean that we live with it and put up with it – we still have to manage those cases of patients who become unwell with it.”
Professor Paul Hunter, from the University of East Anglia, said the way infections were reported might need to change as Covid-19 becomes endemic so that people are not “frightening ourselves” with high numbers “that actually don’t translate into disease burden”.