Family lawyer Wick Sollers in a letter sent Friday to the NCAA said the Paternos would like to appeal the "enormous damage" to Penn State, the community, athletes and the late Hall of Fame coach. He died in January at age 85.
The NCAA did not return messages seeking comment on whether college sports' governing body would consider such an appeal.
The landmark NCAA penalties handed down last month included a four-bowl ban, scholarship cuts and 111 vacated wins from 1998-2011, meaning Paterno no longer has the most coaching victories in major college football.
The family said the NCAA acted hastily and without regard for due process, and that it accepted the results of the school's internal investigation without further review.
The report from former FBI director Louis Freeh said Paterno and three school officials concealed allegations against Sandusky dating back to 1998. Paterno's family and the three officials have all vehemently denied the conclusions.
Once a revered defensive coordinator, Sandusky is awaiting sentencing in jail after being convicted in June of 45 counts of sexually abusing young boys.
Penn State handed the Freeh report to the NCAA, which announced the strict sanctions accepted by Penn State on July 23. Also among the penalties was a $60 million fine.
The school accepted the sanctions and signed off on a consent decree, with President Rodney Erickson saying later he didn't see a better option - the threat of the "death penalty," or the total shutdown of the football program, was hanging over the school.
A Penn State spokesman declined comment Friday on the Paterno family's intent to appeal.
Paterno's lawyer said the family had a right to file an appeal because it was named in the NCAA's consent decree with Penn state, as well as the Freeh report. The family said it hoped to formally submit an appeal and requested oral arguments before the NCAA's infractions appeal committee, its executive committee or other leaders.
"Furthermore, the NCAA and Penn State's Board (of Trustees) Chair and President entirely ignored the fact that the Freeh Report, on which these extraordinary penalties are based, is deeply flawed because it is incomplete, rife with unsupported opinions and unquestionably one-sided," Sollers wrote the NCAA.
Sollers called the sanctions possibly "the most important disciplinary action in the history of NCAA," but that it had been handled fundamentally inappropriate and unprecedented manner.
Michael McCann, director of the Sports Law Institute and at a professor at Vermont Law School, said Friday he doubts that the Paternos do have legal standing to appeal, and that the NCAA is likely to reject it out of hand, if they respond at all.
"He's not alive, and his family itself would not seem to have any legal standing to challenge the NCAA," McCann said, "at least in terms of filing an appeal."
Associated Press writer Mark Scolforo in Harrisburg contributed to this report.