The midfielder played the full 90 minutes as Leverkusen defeated Rangers 1-0 (4-1 on aggregate) to progress to the Europa League quarter-finals, where they will play Inter Milan in Germany on August 10.
The youngster is understood to have told the German club of his desire to join the Blues, who believe they can agree a deal of around £70million.
But while reports have suggested Havertz wants to push through a deal as quickly as possible, Leverkusen have no intention of selling him before their Europa League run ends.
Bosz says the club are doing all they can to protect Havertz from the immense attention he is receiving at this moment, but insists the versatile midfielder is not struggling with the pressure.
"We are watching all this closely," Bosz told DAZN and Goal. "Of 20 interview requests, 18 want to speak with Kai. Then we must make sure he doesn't go 18 times.
"He is an exceptional player, but even exceptional players can have a bad day. If we lose, he is always the first to be criticised, even if he didn't play that badly.
"I often talk to him and explain how I saw his game. Then there are games like the one against Hertha Berlin, when the fans whistled at his substitution.
"I told him that maybe he learned the most in that game, where he was really not good.
"But Kai is an intelligent boy overall and knows how the world works."