Private students – those who are either repeating the class or improving their marks – had been instructed to appear for the exams.
The board has directed that the schools may be inspected in such a manner that at least each category of schools -- private, government, KVS and NVS -- is covered.
Yash Handa, a student of KR Mangalam World School, Gurugram, termed the CBSE's decision as a big relief.
The top court was hearing a plea which has sought directions to states to not hold board examinations in view of the COVID-19 pandemic.
The state government said it will tentatively hold the Class 12 examination in the last week of July and the time table would be issued shortly.
The bench said that if students are given the option of internal assessment then as per the schemes, their results will be declared by July 31 and thereafter they can opt for improvisation.
Calling the policy unfair, they demanded entrance test-based admissions in central universities like Delhi University (DU).
The top court, meanwhile, made clear that there will be no roll back of the decision to cancel class 12 board exams.
A final decision in this regard would be taken on Friday at a meeting between the education department and various other stakeholders.
A section of parents say, the students who are not happy with the scores awarded under the new method should be given an option to take the board exams later.
Nearly 12 lakh students had registered for the examinations, traditionally held in the month of March every year.
The millennial Indians born in the third and fifth years of this century have been the worst sufferers of the pandemic and its related misgovernance.
But a few are not in favour of the exams getting postponed to July-August or the evaluation system either.
The decision was in view of the high prevalence of the second wave of COVID-19 in several parts of Tamil Nadu and also considering the chances of a third wave, the government said.
The school education minister Anbil Mahesh Poyyamozhi, who hails from Tiruchy, has been conducting a series of meetings with academicians, experts and others regarding the conduct of the exams