This is their first break since their trip to Mustique last year.
Government sources told The Sun on Sunday that Mr Johnson was “exhausted”.
Since his last vacation, the Prime Minister has tackled the pandemic, almost died of Covid-19, and welcomed his son Wilf.
However, the break is likely to draw criticism as it comes amid a fuel crisis and truck driver shortage, which has threatened supply chains.
Business Secretary Kwasi Kwarteng defended Mr Johnson over his departure on vacation.
He told Times Radio: “I think he’s gone. I don’t know where he went, but what I would say is that I am in regular contact with him.
“He also had a year and a half in which he almost lost his life due to Covid, his mother passed away very sadly two or three weeks ago and he may have decided to take a short break.
“I think it’s something reasonable.
“I am in regular contact with him on WhatsApp, I spoke to him only a few days ago. I don’t know when he is supposed to have left the country.
Education Secretary Nadhim Zahawi said on Sunday that an accelerated program allowing 5,000 people to be trained as heavy truck drivers will be extended by 2,000 places.
It is hoped that intensive “skills training camps” will increase the number of truck drivers and help end shortages.
However, the free classes lasting up to 16 weeks will not start until next month, meaning they will do nothing to alleviate concerns about shortages over Christmas.
About one in six adults in Britain has been unable to purchase essential food items in the past fortnight, according to the Office for National Statistics (ONS).
The figure was almost as high as a quarter for non-essential food items, with the lack of truck drivers partly responsible for the shortages after Brexit cut the EU’s labor supply.
With the courses being backed by £ 17million, the government is committed to ensuring that successful interns are guaranteed a job interview with a local employer.
But the first fully trained truck drivers will not be “road ready” until February, Downing Street has acknowledged.