If achieved, this will mean the target for offering a first dose to all adults will be hit more than a month early, as the original date was the end of July.
Many over-18s have already begun to receive appointment notifications by text and email, Ms Sturgeon told MSPs on Tuesday.
The remainder will receive a letter over the course of next week, she said.
The First Minister said the gap between first and second doses has been shortened, from 12 to eight weeks.
This means all adults should have been given two doses by around the end of September, although the First Minister did not give a target date for this.
Ms Sturgeon told Parliament: “We have set the target to offer everybody in the adult population a first dose by the end of July.
“In actual fact, by the end of next week, everybody in the adult population will have been offered their appointment for the first dose, in fact many of the 18 to 29-year-old age group will be getting those appointments by text and by email already.
“But letters to the remainder will go out over the course of next week, so appointments will be scheduled for the entirety of the adult population that haven't already had the first dose by the end of next week.
“Second doses we are seeking to do within eight weeks now of the first dose, so you can work from that what the outer limit is in terms of seeking to get the adult population vaccinated with a second dose.
“Obviously supplies remain our biggest constraint, so all of that is subject to sufficient supplies, but we are doing that as quickly as supplies allow.”
Most recent figures from Public Health Scotland show 79 per cent of adults have so far been given a first dose of vaccine.
This includes 30 per cent of those in the 18 to 29-year-old age group.
Downing Street earlier today confirmed all over-18s in England would be offered a first dose of a coronavirus vaccine by the end of this week.