The 79-year-old arrived in Athens on Wednesday evening to visit his mountain villa, and shared a video of his plane landing in the capital on social media platform Instagram.
He also posted a picture of himself wearing a face mask, which appeared to be taken in an airport.
Mr Johnson flew to Athens via Bulgaria, the Daily Mail said, because the Greek government has banned direct flights from the UK until mid-July.
He told the paper he was visiting the country on “essential business trying to Covid-proof my property in view of the upcoming letting season”.
“I need to set up distancing measures at the property because they’re taking it very seriously here. The Greeks are trying to stop bulk arrivals from the UK but they were quite happy to have me coming in.
“All they wanted to know was where I was coming from and what I was doing. Then I had my temperature taken and was swabbed twice.”
Mr Johnson said air bridges must be set up “as soon as possible”, adding: “From what I’ve seen the arrival of the British will not be a danger to the Greeks because they’re so careful here.”
But his trip has angered MPs, with shadow minister Rosena Allin-Khan tweeting: “Most people have been following the guidelines and socially distancing – not everyone will get a holiday this year.
“Those closest to the Prime Minister have different rules though.”
Labour MP Kate Osborne said: “Dominic Cummings broke the lockdown rules now it’s @BorisJohnson’s father.
“One rule for the elite and another for everyone else it seems.
“They’re laughing directly in the face of people up and down this country. Shame.”
And Labour’s Neil Coyle tweeted: “Junior Johnson axes freedom of movement, undermining rights for 60 million Brits.
“Senior Johnson sneaks to his Greek villa during a global pandemic.”
Liberal Democrat Alistair Carmichael said: “This is simply further evidence that when it comes to following the rules, it is one rule for the Conservatives and one rule for everyone else.
“Whilst the Prime Minister is continuing to ask people to make huge sacrifices, especially the people of Leicester, he must reflect on how much of a kick in the teeth these reports will be.”
The Foreign Office currently “advises British nationals against all but essential international travel” because of the coronavirus pandemic.
The Prime Minister’s official spokesman confirmed that the Foreign Office advice remains in place, but said: “It is for individuals to make the judgments themselves.”