Reports reaching Pulselive.co.ke newsdesk indicate that Ndemo Nyagarama, 44, died on Monday evening while receiving treatment at the hospital.
A family member divulged that the Governor's firstborn son had been admitted to the medical facility for two months.
"He has been unwell and we are grateful to the Nairobi family for taking care of him. Unfortunately, he has left us," Momanyi Nyagarama was quoted by the Nation as saying.
The son is said to have been suffering from meningitis.
The Orange Democratic Movement (ODM) Party sent its condolences to the family.
"We have received with profound shock the passing on of Ndemo Nyagarama, son to H.E. John Nyagarama, Gvn. Nyamira County this evening at the Nairobi Hospital due to acute meningitis.
"The ODM family condoles with the family of Hon. Nyagarama & prays for God's peace during this time," the message read.
The death comes at a time when the Governor is also in the US where he has been receiving treatment for one month now.