The bills' sponsors - state Senator John Tassoni Jr. and state Representative Joseph McNamara - say the rules have changed from the old days of school yard bullies shaking down classmates for lunch money.
They say cyberbullying - defined as textual, verbal, or graphic harassment transmitted by computer, cellphone, telephone, or other electronic devices - is on the rise.
Under the bill, repeated violations that threaten the physical or emotional well-being of any student would be grounds for filing a complaint with the Family Court to have the student engaged in the alleged bullying deemed delinquent.
The problem of cyberbullying is getting increased attention nationwide.
Last month, Facebook, the world's second-largest social networking website, announced it was adding more than 40 new safeguards to protect young users from sexual predators and cyberbullies in an agreement with attorneys general from several states.
And last week in Los Angeles, a Missouri woman pleaded not guilty in federal court to charges in an Internet hoax blamed for a 13-year-old girl's suicide.
Investigators say Lori Drew, 49, helped create a MySpace account that appeared to belong to a 16-year-old boy who did not exist and used the fake account to send cruel messages to 13-year-old neighbor Megan Meier, including one stating the world would be better off without her. Megan hanged herself in 2006.
A second bill would make "cyberharassment" a crime.
A first offense would be treated as a misdemeanor and subject to a $500 fine. Second and subsequent offenses would be treated as felonies with fines up to $6,000 or two years in prison.
Both bills now head to Governor Don Carcieri's desk.