The Prime Minister did not confirm when the travel rules will be eased, but it is likely to happen before the February half-term break.
This will be a major boost for travel firms and families planning an overseas trip.
On a visit to Milton Keynes Hospital on Monday, Mr Johnson said: “What we’re doing on travel, to show that this country is open for business, open for travellers, you will see changes so that people arriving no longer have to take tests if they have been vaccinated, if they have been double vaccinated.”
Industry body Airlines UK said it was a “landmark day”.
The announcement only applies to England, but the devolved administrations in Scotland, Wales and Northern Ireland have recently implemented Westminster’s changes to international travel rules.
Fully-vaccinated arrivals must currently pre-book and take a post-arrival test from a private supplier.
This can be a lateral flow test, which typically costs around £19.
It is understood further changes mean fully-vaccinated arrivals will only have to complete a simplified version of the passenger locator form.
It is also expected vaccines used by more countries will be recognised for travel, and the rules for travellers who are not fully vaccinated will be eased.
Arrivals who are not fully vaccinated must currently take a pre-departure test and two post-arrival PCR tests, which are more expensive than the lateral flow version.
They must also self-isolate for ten days.
The decision to ease travel rules follows growing pressure from the sector, which claims restrictions are ineffective.
EasyJet boss Johan Lundgren welcomed the update, saying “millions of our customers” will be “delighted to see the return of restriction-free travel in the UK”.
He went on: “We believe testing for travel should now firmly become a thing of the past.
“It is clear travel restrictions did not materially slow the spread of Omicron in the UK and so it is important that there are no more knee jerk reactions to future variants.”
Mr Lundgren said the airline intends to return to “near-2019 levels of flying this summer”.
Airlines UK boss Tim Alderslade said: “This is a landmark day for passengers, businesses and UK plc.
“Nearly two years since the initial Covid restrictions were introduced, today’s announcement brings international travel towards near-normality for the fully vaccinated, and at last into line with hospitality and the domestic economy.”
Christophe Mathieu, chief executive of Brittany Ferries, said bookings for spring and summer holidays were already “roaring ahead”, but the announcement means “we can expect demand to soar for the February half-term”.
Abby Penlington, director at ferry trade association Discover Ferries, said the announcement will make travelling “easier, cheaper and will be a further boost to consumer confidence”.
UK transport secretary Grant Shapps will announce details of the new travel regulations in a statement in the House of Commons on Monday afternoon, following a meeting earlier in the day of the Cabinet’s Covid-19 operations committee.