The Daily Beast reported the suits would be filed against the university over administrators' failure to take action after learning of the alleged sexual abuse.
"This may be the most high-profile sexual abuse case ever," Harrisburg, Pa., lawyer Benjamin Andreozzi said.
Andreozzi is representing at least one of the eight alleged victims included in grand jury charges and said other victims may step forward when they realize they're "not alone."
A second lawyer, requesting anonymity, told The Daily Best he was representing another alleged victim.
Meanwhile, a state police source told the publication phone lines set up for victims and tips have been inundated with calls.
Sandusky, 67, has been charged with 40 counts of abusing at least eight boys in a 15-year span, all of whom he met through The Second Mile, the non-profit charity he founded. At least one more victim is believed to have come forward since, The Daily Beast said.
ABC News reported Andreozzi said a suit could include officials at Penn State and The Second Mile who did not report alleged assaults.
Newly appointed Penn State President Rodney Erickson pledged the university's commitment to victims and to increasing awareness of child sexual abuse.
"We understand there will be lawsuits filed. We're prepared to do the right thing for all the victims," he said on "Good Morning America."
"We will do everything we can do. … We're going to engage in a wide range of programming that will raise the issue of child sex abuse, to make this a national issue."