Descartes supposed have said "I think therefore I am". For him thinking entity is the Self. While Hindu sankhya-yoga system puts, both mind and intellect in products of prakriti/pradhana, separate from self that is purusha. Vedanta schools also seems to subscribe to samkhyan model of thinking entity not being self and self being only a witness/observer. Do any of the other darshanas consider thinking entity as Self or do any of the non darshana Hindu philosophical schools consider thinking entity as Self?
Though I don't know of any specific sources which directly claim that Self is Intellect/Thought, ie Atma is Buddhi, but you can certainly find that such schools existed - one being the Buddhist - as their arguments have been mentioned and systematically rejected in the Dakshinamurthy Stotra here.
One needs to be careful when comparing with Descartes (or any Western philosopher) though, as he doesn't differentiate the Mind, ie mental substance, into Mana, Buddhi, Ahankara, etc as is typically done in Indian system.
So, the 'Mind' of Descrates could also mean Prana(Life-Breath) or Vijnana(Consciousness) or Buddhi(Intellect) or Ahankara(Ego), and even the Carvakas could be considered here.