Fee-paying schools such as Fettes College in Edinburgh, the High School of Glasgow, and Glenalmond College in Perth and Kinross will be assessed over the next two years.
Loretto School in Musselburgh, Robert Gordon's College in Aberdeen, Wellington School in Ayr and Edinburgh Academy are also on the target list.
The aim is to maintain public confidence in charitable status, according to the Office of the Scottish Charity Regulator (OSCR).
Each school will have to confirm the accuracy of information already held by OSCR and provide further evidence in order for a full assessment to be made.
OSCR staff will then consider whether the schools meet the charity test set out in the Charities and Trustee Investment (Scotland) Act 2005.
The review is part of the watchdog's overall programme of reviews of charitable status, focusing on "priority groups" whose status may be uncertain.
David Robb, the watchdog's chief executive, said: "Charities need to be seen to be providing public benefit. That is what the legislation requires the regulator to ensure.
"Our reviews of charitable status have been ongoing since 2006 and we have gained considerable experience as Scotland's charity regulator. We believe it's right, as agreed by our board in January of this year, that we now proceed to complete our assessment of the remaining schools in this priority group.
"We have reviewed 13 schools so far, to reassure the public that the requirements of charitable status are protected. We look forward to engaging in dialogue with these charities so that they can demonstrate the benefit they provide and how they address any restrictions."
Of the 13 schools reviewed so far, four private schools were issued with directions in 2008 to take steps to meet the test. The schools were given a year to make changes such as widening their bursary schemes and improving the chances for pupils from poorer backgrounds to enrol.
They all ultimately met the test. Failing to do so risks being struck off the charitable register, meaning the schools would lose out on thousands of pounds-worth of tax breaks.
Seven other independent schools, including Prince Charles's old school, Gordonstoun, passed the charity test.
John Edward, director of the Scottish Council of Independent Schools, said: "All 13 independent schools that have been reviewed to date have passed the charity test and demonstrated that they provide public benefit, and we're confident that the other schools are well placed to do so too.
"Today's announcement of the next 40 schools to be reviewed is simply part of OSCR's ongoing review of registered charities which has been in place since 2006, following the introduction of Scottish Parliament legislation."
Using the same test, the watchdog has granted charitable status to around 5,000 new charities. The assessment looks at specific factors such as whether public benefit is provided.
OSCR has also identified several special and grant-aided schools that are being reviewed by other bodies. It said it will assess these schools' charitable status at a later date.
OSCR was established six years ago under the Act and oversees around 23,500 charities. It is a non-ministerial department based in Dundee, reporting directly to the Scottish Parliament.
The list in full:
Aberdeen Waldorf School
Belhaven Hill School Trust
Clifton Hall School
Governors of Dollar Academy Trust
Edinburgh Steiner School Trust
The International School of Aberdeen Educational Trust
Kelvinside Academy War Memorial Trust
Kilgraston School Trust
Moray Steiner School
Morrison's Academy Trust
Queen Victoria School
Robert Gordon's College
St Columba's School
St Georges School for Girls
St Mary's School, Melrose
Glasgow Academicals War Memorial Trust
The High School of Glasgow
Proprietors of the Edinburgh Academy
Governors of the Fettes Trust
Edinburgh Merchant Company Education Board
Seamab Learning and Care Services
Struthers Memorial Church
Lewis Independent Christian School, Stornoway
Mannafields Christian School, Edinburgh
Oakwood School, Laurieston
Springvale School, Bridge of Don, Aberdeen
Contextual targeting label: