The Hoops qualified for the last 16, after a stunning group stage that saw them beat La Liga giants Barcelona, and the Israel international is aiming for more European success
Neil Lennon’s men shocked many by progressing from a group containing Barcelona, Benfica and Spartak Moscow, securing a memorable win over the Spanish side at Parkhead.
"We are still hungry to prove a lot in Europe," Kayal told reporters. "Two months ago, no one believed we could make it to the last 16.
"The manager changed the mentality for the players. It is a young squad and nobody wants to stop now. We are looking forward to more success and to taking the team forward.
"We don't think the last 16 is the maximum possible for this team. I don't agree with the people who say it is. Maybe we can surprise them again."
The SPL leaders are not looking to spring another shock on one of the competition's giants, with Kayal naming a lesser name as the Hoops' ideal opponent in the next round.
"I think Malaga would be a brilliant draw for us," he continued. "We need to do the job in the Champions League. We want to believe we can win against anyone, because we are professionals. We are playing on a higher level now.
"There is no one we want to avoid in the draw. We would be happy to take Malaga but we just need to wait until Thursday to see."
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"The SPL is very different from the Champions League and we are trying to change the mentality, to make sure we also win games in the SPL. It is not easy to change from one to the other," Kayal said.
"I think we are in a good position now, four points clear with a game in hand. I think it is possible we can have a big advantage at the top of the SPL by the time the next Champions League game comes around.
"We try to do the maximum in the league, it's not easy to play in four competitions.
"But we have a big squad and we will try to do the maximum in the SPL now, because we know it is important for us if we want to be in the Champions League again next season."