Kumar spoke to Mukherjee over telephone and conveyed his congratulations to him for his election as the next president.
Kumar requested Mukherjee to undertake a visit to Bihar shortly.
Kumar said Mukherjee accepted the invitation with humility.
The Chief Minister said his government would send a formal invitation to Mukherjee for visiting Bihar after July 25.
The JD(U), despite being a part of the NDA, had extended support to Mukherjee in the presidential poll in preference to BJP-backed candidate, P A Sangma.