Rahul has not played competitive cricket since May but was picked for the Asia Cup after recovering from a thigh injury that had kept him out of action.
He then suffered a fitness issue that was not linked to his original injury during his rehabilitation at the National Cricket Academy (NCA) in Bengaluru.
"KL has had a good week with us, playing well, progressing really well but he will be unavailable for the first part for the Kandy leg of the trip," Dravid told reporters.
"The NCA will be looking after him for the next few days while we are travelling. Signs are looking good, and he's progressing really well.
"We'll assess him on the fourth (of September) and we'll take it from there."
In Rahul's absence, Ishan Kishan is set to don the wicketkeeper's gloves in India's opener against Pakistan on Saturday and against Nepal two days later.
India have another wicketkeeping option in Sanju Samson, who will travel with the side as a reserve player.
The Asia Cup gets underway on Wednesday with co-hosts Pakistan taking on Nepal in Multan.
India have refused to tour Pakistan because of the soured political relations between the neighbours and will play their matches in Sri Lanka instead.
