"I was really impressed with Arshdeep, how he's come through," Kumble said on ESPNcricinfo's Open Mic programme. "I worked with him for three years and I could see the kind of development that he has had in the T20 format, and last year's IPL was a classic example of how he handled the pressure.
"He perhaps bowled the tough overs for the team and yeah, you don't always look at the wickets column in the T20 game, you look at what moments the bowler comes up with. And the temperament that he's shown, it's wonderful. We saw that again in the India-Pakistan game. When you have 90,000 people at the MCG, it's always challenging.
"So yes, Arshdeep has certainly matured and I'd like to see him carry on. Probably what Zak [Zaheer Khan] did for India, I expect Arshdeep to go on to do some wonderful things for India."
"Jitesh is someone I rate really, really highly," Kumble said. "He's a wonderful cricketer. He's very mature. He has done all the hard yards in domestic cricket. He can hit the ball, not just off the spinners but off the fast bowlers as well. He is someone who can hit off the back foot and he is someone who I think can do that finisher's role for India. If not now, he should be looked at post this World Cup when the selectors sit down to look at the next lot of cricketers, I'm sure Jitesh's name will already be there.
"He does a finisher's role. My take on T20 cricket is about specific roles that they play for their teams. And Jitesh, I know he started off as an opener, but he's currently in a finisher's role. He bats at No. 5-No. 6. And that's the kind of thing you need to do. If you look at stats, then you'll pick up the top two batters which are generally the openers or No. 3. They end up getting all the numbers in T20 cricket, but Jitesh, in his role has done exceptionally well."
When asked which team had the best bowling attack in the ongoing men's T20 World Cup, Kumble picked Pakistan, who have Shaheen Shah Afridi, Haris Rauf and Naseem Shah as the frontline quicks.
"Pakistan certainly have the best pace-bowling attack, for sure," Kumble said. "They lacked the allrounder that Australia have. Australia have, overall, a good attack, in terms of having options. India certainly has good spinners. If you ask me, I think Pakistan have the best fast-bowling attack."