The English side have hired Kumble as one of their overseas players and was due to arrive in London on Tuesday.
Surrey have gone to the top of the second division this season on the back of 68 wickets taken by spinners Ian Salisbury and Nayan Doshi.
"Nine months ago it was a no-brainer," coach Alan Butcher told The Times.
"But the two lads have done really well. Anyway, there is nothing in the rule book that says you can't play three spinners in the same side and that's what we are planning to do at Guildford (in their next fixture) this week."