"Shehzad has breached the central contract in which it is clearly written that you will not involve in any religious or political discussion with others," Khan said in the southern city of Karachi.
"There can be a disciplinary action against him."
The PCB has already formed a three-member committee headed by international cricket operations director Zakir Khan to investigate the issue.
Shehzad was caught on camera last Saturday telling Dilshan "if you are a non-Muslim and you turn Muslim, no matter whatever you do in your life, straight to heaven."
Dilshan's reply was not audible, and Shehzad went on to say, "Then be ready for the fire."
They were walking back to the dressing room after Sri Lanka won the third and deciding one-day international in which Dilshan scored an unbeaten 50.
There was no official complaint from Sri Lanka Cricket or match officials, but Shehzad told PCB officials on Wednesday it was a "personal discussion," with Dilshan and there was nothing more to it.
Dilshan told espncricinfo.com that he would also not lodge any complaint.
"I don't remember even what I said to him," Dilshan said. "I have no issues at all. I was just quite happy with the win at the time."
Dilshan was born to a Muslim father and a Buddhist mother and was known as Tuwan Mohamed Dilshan.
He shed his Muslim name shortly after his international debut in 1999 and took up the name Tillakaratne Mudiyanselage Dilshan with an ethnic Sinhalese-Buddhist identity.
Shehzad is due to accompany the Lahore Lions to India to compete in the Champions League next week, and is one of the top contenders to be named Pakistan's Twenty20 captain.