More than 17 million people have been given a jab since the UK's Covid vaccine rollout began in December 2020.
But Boris Johnson said he now wants the programme to "go further and faster".
He said the July target would allow vulnerable people to be protected "sooner" and would help to further ease lockdown rules across the country.
NHS England chief executive Sir Simon Stevens said there were "early signs" the vaccine rollout was contributing to a fall in hospital admissions.
The PM is due to hold a final meeting with senior ministers about how to ease England's lockdown later, before he sets out the full "road map" on Monday.