The First Minister appeared to announce the visit on Twitter and urged the Prime Minister to meet with her and discuss “working together”.
Ms Sturgeon has now written to the Prime Minister to invite him to Bute House in Edinburgh.
She tweeted: “I understand the PM will visit Scotland later this week. Since this would be our first opportunity to meet in person for a while, I’ve invited him to Bute House to discuss Covid/recovery.
"We differ politically, but our governments must work together where we can.”
Writing to Mr Johnson, she said: "I understand you will be in Scotland later this week and thought this might offer us an opportunity to meet in person in Edinburgh for a discussion on the current Covid sitation and our respective plans on recovery- focussing, obviously, on the areas where it is important that our governments work together.
"I would be happy to welcome you to Bute House and hereby extend an invitation.
"If you are agreeable, perhaps your Private Office could contact mine to discuss suitable arrangements?”.
News of the visit came as a surprise to MPs on both sides of the Chamber. The SNP’s John Nicolson tweeting that it was “extraordinary” that the First Minister had to learn “indirectly that Boris Johnson will be in Scotland”, while Scottish Tory MP Andrew Bowie criticised the First Minister for apparently revealing the visit.
He said: “Pretty poor show, utterly unprofessional (although par for the course with our First Minister) and against all standard security considerations if the PM is coming North to announce it on twitter in advance.
“Even if he is, he's not "visiting", he's Scotland's Prime Minister.”
The Prime Minister has not visited Scotland since late January when he visited a Lighthouse Laboratory despite Ms Sturgeon criticising the trip as “non-essential”.
At the time the official SNP account posted a picture of Mr Johnson getting off a plane in Scotland with the caption: "Stay at home. Protect the NHS. Save lives."
His last visit to Bute House saw the Prime Minister sneak out of the back door after being faced by pro-indy and anti-Brexit demonstrators.
Last week the Chancellor Rishi Sunak visited Scotland to discuss Scottish tourism figures and the UK Government’s “Scottish plan for jobs”.
Downing Street did not respond to requests for comment on the Prime Minister’s trip.
It came as it was reported yesterday Police Scotland had to abandon a proposed codename for its mission to protect Boris Johnson during his next visit north of the border, amid fears it would offend him.
Sources inside the force told The Sun that the mission was initially entitled ‘Operation Bunter’ - which some feared could draw unflattering comparisons with Billy Bunter, a pupil at the fictional Greyfriar’s School made famous in the 20th century children’s stories written by Charles Hamilton.
Bunter, depicted in illustrations - and in a 1950s television adaptation - as an overweight teenager, is regularly portrayed as greedy, lazy and deceitful.
Instead, ‘Operation Aeration’ was chosen to avoid “some sort of diplomatic incident,” they added.
A Police Scotland spokesperson said: “Operational names are auto-generated by computer and can be changed if deemed to be inappropriate.”