The news comes as on Monday cafes, restaurants and beer gardens will open, along with shops, gyms, libraries and museums.
Further lockdown easing is due to take place on May 17, with Scotland planning to move to Level 2 of its restrictions.
However, speaking at the coronavirus update on Tuesday, Ms Sturgeon added: "If circumstances permit, then I can confirm that it would be the intention to move to level one on Monday June 7 then moving to level zero in late June.
“Then as we go into the deeper part of the Summer, something much more like normality over the course of July.”
If the plan goes ahead, by June 7, eight people from up to three households would be able to socialise indoors in a public space and six people from up to three households in a private dwelling.
Hospitality would remain open indoors until 11pm and 12 people from 12 households would be able socialise outdoors.
From late June, ten people from up to four households would be able to socialise indoors in a public place and eight people from up to four households in a private dwelling.
Up to 15 people from up to 15 households can socialise outdoors.
All sport activities would be permitted.
The news comes as two deaths from coronavirus and 178 positive tests have been recorded in the past 24 hours.
This continued decline in the number of cases means the country could relax restrictions, the First Minister said.
It brings the death toll under this measure – of people who first tested positive for the virus within the previous 28 days – to 7,644.
Giving a Covid-19 update, the First Minister said 224,092 people have now tested positive in Scotland, up from 223,914 the previous day.
The daily test positivity rate is 1.4%, down from 2.5%.
There are 106 people in hospital confirmed to have the virus, up two in 24 hours, and 13 patients are in intensive care, down one.
Ms Sturgeon said 2,750,052 people have received their first dose of a coronavirus vaccine, up 2,358 from yesterday, and 797,267 have received their second do