The legal notice was sent to them yesterday by the lawyer of Uday Kant Mishra, who is also a long-time friend of the chief minister.
The RJD has, however, denied getting any notice from Mr Mishra as yet.
Mr Mishra's lawyer Vinay Shankar Dubey today said Prasad and Tejashwi have been asked to reply to the notice within 15 days.
They should tender a public apology for levelling personal allegations, failing which a case would be registered against them under relevant sections of the law, Mr Dubey said.
RJD spokesman Shakti Singh Yadav said the notice has not reached them and they will take appropriate action after receiving it.
Mr Dubey said the "objectionable comments" were made by Mr Yadav against his client in the context of the multi-crore Srijan scam.