Video report by ITV News Political Correspondent Romilly Weeks
Boris Johnson warned: "It's worth stressing that on the continent right now, you can see, sadly, there is a third wave underway.
"People in this country should be under no illusions that previous experience has taught us that when a wave hits our friends [in Europe], I'm afraid it washes up on our shores as well."
"I suspect we will feel those affects in due course," he added, stressing the importance of pushing on with the vaccination campaign to try to limit the impact of more infections.
His comments come as the European Union ramped up its rhetoric on having the power to block exports of the Covid vaccine.
The EU's vaccination campaign is lagging behind the UK's, with the bloc claiming pharmaceutical giant AstraZeneca has "underproduced and underdelivered" on its contract.
Speaking at BAE Systems in Preston, Mr Johnson said: "I'm reassured by talking to European partners over the past few months that they don't want to see blockades.
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"I've talked to our friends repeatedly, we're all facing the same pandemic, we're all facing the same problems.
"A vaccination campaign, developing vaccines, rolling them out, these are international projects and they require international cooperation".
On Monday, a spokesperson for EU Commission president Ursula von der Leyen insisted it was not set on "banning vaccine exports" but rather wanted pharmaceutical firms to meet their contractual obligations on the delivery of jabs.
The PM's warning comes after several regions in France, including Paris, spent their first weekend under a limited month-long lockdown amid soaring cases.
Labour’s shadow home secretary Nick Thomas-Symonds responded by calling for a "more comprehensive hotel quarantine system", and said it was "unacceptable" that an enforced hotel stay applied to only 1% of international arrivals.
The coronavirus situation in Europe has prompted ministers to manage expectations for Brits eyeing up a holiday abroad.
Care minister Helen Whately said on Monday it was "premature" to consider booking a holiday abroad with Covid rates on the rise elsewhere.
She urged Britons to "hold off" on making foreign travel plans.
Defence Secretary Ben Wallace said on Sunday UK tourists "should not forget that just across the channel" in reference to growing infections.
"That's got to be worrying, because what we don't want is all our hard work...is undermined by a new variant getting imported, because we lifted it too quickly."
"I haven't booked a holiday and I would recommend that as much as the temptation is massive to want to have a holiday and we all want a holiday after this, first and foremost, let's see what we get in April, May and we can reflect on it."
Mr Johnson faces another battle this week too, as he prepares to ask MPs to extend coronavirus restriction measures for a further six months.
Ministers say the extension would ensure furlough can continue to apply even after all measures have possibly been scrapped.
The PM's press secretary Allegra Stratton said: "As you would expect, the Prime Minister is using every opportunity when he meets his Conservative MPs to make the case."
But the PM will need to convince his own MPs, including the powerful backbench 1922 Committee.
Critics, including former chief whip Mark Harper, have warned the continuation of restrictions until June 21 at the earliest under the road map is draconian because the success of the vaccination programme means they are not needed.
Critics, including former chief whip Mark Harper, have warned the continuation of restrictions is “draconian” because the success of the vaccination programme means they are not needed.
Ms Stratton said Mr Johnson believed "this is a cautious but – if we do it right – irreversible road map" which would take time because of the need to assess the impact of each measure.
The government’s plan would see shops and pub beer gardens open on April 12 at the earliest, with indoor hospitality allowed from May 17 if all goes to plan and the final stages of lockdown being lifted on June 21.