Though Mr Kumar did not name anyone, his remarks riled Tejashwi, who is the son of Lalu Prasad, and he took to Twitter daring the chief minister to swear on an affidavit that his son will never enter politics.
"If Nitish uncle is left with even an iota of morality and integrity, he should sign a sworn affidavit stating that his son will never enter politics," Mr Tejashwi tweeted.
Mr Tejashwi (28) is the younger son of RJD chief Lalu Prasad and often refers Mr Kumar as "chacha" (uncle).
Mr Kumar, who is also the national president of the Janta Dal (United), was addressing a function organised by the party's youth wing here.
"The role of youth in politics is very important. We learnt our ropes while we were students and I feel glad when I see youngsters taking an interest in politics. But looking around, we see so many young leaders who owe their position in politics to their families, basically they are interested in acquiring a post in order to make money. A far cry from our times," he said taking a veiled dig at Mr Tejashwi.
Mr Kumar had resigned from his post last year following the refusal of the RJD, which he was then having an alliance with, to pay heed to his concerns about corruption cases against Tejashwi Yadav, who was then his deputy.
The JDU chief later formed a new government with the BJP, which came forward with an offer of unconditional support.
"I am delighted that this Yuva Sankalp (youth resolve) conference is being held on a day when we celebrate the anniversary of Sampoorna Kranti the call for a total revolution by Loknayak Jaya Prakash Narayan, who had inspired many youngsters like myself to devote themselves to public life," Mr Kumar said.
Mr Kumar, Lalu Prasad and Deputy Chief Minister Sushil Kumar Modi are products of the JP movement in which they participated actively as student leaders of the Patna University.
"Care must be taken not to get trapped in the prevailing culture of empty talk. As you all know, there are political figures who excessively use social media and go on tweeting about things they do not even understand. They talk big but have no idea of what they stand for," Mr Kumar said alluding to Tejashwi Yadav.
The chief minister, who has been accused by the opposition of going soft on the BJP when many of its workers were allegedly involved in incidents of communal tension earlier this month, said, "Intellectuals often ask me why I do not speak on the prevailing social strife. They fail to see that I keep doing that, focusing on issues like social reform which alone can ensure enduring peace."
Mr Kumar also exhorted party cadres to apprise the common people of his governments measures like prohibition and campaign against dowry and child marriage.
"Today is the World Environment Day. Our party has been committed to environment conservation too as is evident from our drive for plantation of one crore trees across the state that was launched a few years ago.
"Besides our government is also making efforts to increase the forest cover in Bihar. We have achieved some success but we want to accomplish more. Please sensitise the people about his issue as well," he added.
Continuing to vent ire on the CM, the opposition leader further tweeted, "In a 60-minute speech, Nitish uncle spends 45 minutes speaking about Tejashwi. Thanks to the uncle for his focussed concentration on the nephew. After all, Tejashwi is just a kid."
A few months ago, the chief minister had remarked that Tejashwi was a kid (bachcha hai) which the young RJD leader took exception to, alleging that Mr Kumar had spoken of him in a condescending manner.