Lawyers for a private equity investor and a former casino executive facing federal prison in the college admissions scandal known as Operation Varsity Blues filed appeals on Monday seeking to have their convictions overturned.

Both men were accused of making payments to have their children admitted to elite universities as athletic recruits, even though prosecutors charged that they lacked qualifications to play Division 1 sports.

The men, John B. Wilson and Gamal Abdelaziz, face the longest sentences yet imposed on parents in the admissions scandal, in which more than 50 parents and college coaches were prosecuted for conspiring with William Singer, a college admissions counselor, to arrange “side door” admissions, primarily by using slots on athletic teams.

Mr. Wilson and Mr. Abdelaziz make similar arguments in their appeals — that donations to universities in an effort to secure admissions are commonplace and do not constitute bribery.

Mr. Wilson, a former business executive, was convicted in October on bribery charges and sentenced to 15 months in prison. He was accused of agreeing to pay more than $1.5 million to have his three children admitted to the University of Southern California, Harvard and Stanford.

Lawyers for Mr. Wilson, 62, of Lynnfield, Mass., say in court papers that the key claim against him — that he paid $220,000 to bribe his son’s way into a spot on U.S.C.’s water polo team in 2014 — is legally flawed.

None of the money was intended to personally enrich anyone at the school, they wrote in court papers.

“Donating to a university is not bribing its employees; the school cannot be both the victim of the scheme and its beneficiary,” said the filing in the U.S. Court of Appeals in Boston by lawyers for Mr. Wilson, including Noel J. Francisco, the former U.S. solicitor general.

Of the total $220,000, Mr. Singer forwarded $100,000 as a donation to U.S.C.’s water polo team, for which Mr. Wilson received a thank-you note. Another $100,000 went to Mr. Singer’s nonprofit foundation, which Mr. Wilson thought would benefit U.S.C., according to the appeal.

Mr. Wilson’s son, who had played water polo in high school, was admitted to U.S.C. based on exaggerated athletic credentials, according to prosecutors.

But Mr. Wilson’s lawyers argue that similar practices were commonplace at the university.

“The court excluded evidence showing that U.S.C. regularly dressed up donors’ children as athletic recruits, including for practice-only or non-athletic team-support roles,” the appeal said.

All the while, the appeal argued, Mr. Singer was pitching his arrangement to Mr. Wilson as common practice. In one taped conversation, he told Mr. Wilson that he was arranging “730 of these side-door deals at 50 or 60 schools.”

Mr. Abdelaziz, 65, of Las Vegas, was convicted of bribery and fraud in connection with his efforts to secure his daughter’s admission to U.S.C. in 2017, purportedly to play basketball, despite the fact that she did not make her high school’s varsity team. He was sentenced in February to a year and one day in prison.

In court papers, lawyers for Mr. Abdelaziz argue that he had a longstanding relationship with Mr. Singer, who had assisted his sons with their legitimate college admissions process.

Mr. Singer told him he could help his daughter get into U.S.C. if Mr. Abdelaziz made a donation to its athletic department and his daughter went through the athletic admissions process as a basketball practice player or team manager.

The donation was intended for U.S.C., not any individual at the school, according to court papers. Lauren Janke, a former U.S.C. coach who created a phony athletic profile for Mr. Abdelaziz’s daughter, has also pleaded guilty in the scheme.

In addition to the payments to U.S.C., Mr. Wilson agreed in 2018 to pay Mr. Singer $1.5 million to have his twin daughters admitted to Harvard and Stanford as purported Division 1 sailing recruits, but they never were enrolled.

By that time, Mr. Singer’s conversations with Mr. Wilson and others were being monitored by federal agents in the wide-ranging investigation.

In addition to counts of bribery and fraud, Mr. Wilson also was convicted of filing a false tax return because he deducted part of the payments to Mr. Singer as a business expense.

U.S.C.’s decorated water polo coach, Jovan Vavic, also was convicted this year in connection with the Varsity Blues investigation, which had earlier ensnared the actors Lori Loughlin and Felicity Huffman, who both chose to plead guilty rather than take their chances with a jury.