Kumar, whose party JD(U)'s support will be crucial for the passage of the bill, met Finance Minister P Chidambaram here to convey the concerns.
"We support Food Security Bill (but)... We have some concerns. We want to implement it but we are concerned about additional financial burden on states," he said after meeting Chidambaram.
JD(U) has 20 members in the 543-member Lok Sabha and its support will be crucial for the passage of the bill.
The proposed legislation promises to give right to the country's 80 crore people to get 5 kg of foodgrains every month at Rs 1-3 per kg.
The Bill has support of many parties but Tamil Nadu Chief Minister J Jayalalithaa has expressed concerns over it saying "there are several flaws".
She had suggested various changes in Bill which sought to replace the Ordinance promulgated by President Pranab Mukherjee earlier.