69-year-old Kovind, who had been a two-time Rajya Sabha member between 1994 and 2006, hails from Kanpur in Uttar Pradesh.
His appointment comes barely few months ahead of the Assembly elections due in Bihar. West Bengal Governor Keshri Nath Tripathi was holding additional charge of Bihar.
A lawyer by profession, Kovind had also been BJP Scheduled Caste Morcha chief.
The appointments will take effect from the dates the incumbents assume charge of their respective offices, it said.
Rajasthan Governor Kalyan Singh was holding the additional charge of Himachal Pradesh.