The invitation came from Cavaliers owner Dan Gilbert and team CEO Len Komoroski, Kosar told ESPN's Mark Schwarz.
Kosar played nine years for the Browns, who drafted the University of Miami quarterback in the 1985 supplemental draft.
He led the Browns to a 12-4 record in just his second season, when the team lost to the Denver Broncos in the 1986 AFC Championship Game.
Kosar was named to the Pro Bowl for the only time in 1987, when he completed 62 percent of his passes and had 22 touchdowns while leading the Browns to another AFC title game, which they again lost to the Broncos.
Kosar never got to have a victory parade, but will get his chance to partake in the one LeBron James had long dreamed of bringing to Cleveland.
Hundreds of thousands of fans are expected to line downtown streets to celebrate James and the Cavaliers winning the NBA title and giving the city its first major professional sports championship since 1964.
The parade is scheduled to start at 11 a.m. in the plaza next to Quicken Loans Arena, which was the epicenter of Sunday night's massive party when the Cavs completed their historic comeback by rallying from a 3-1 deficit to stun the Golden State Warriors in the seven-game series. Cleveland is the first team in the NBA Finals to come back from that far down.
James, who delivered on his promise to win a title for his home state in his second year back with the Cavs after four years with the Heat, will ride with teammates on floats. The procession will leave "The Q" and travel south past Progressive Field, home of the Indians, before heading down Carnegie Avenue. The 60-unit parade with floats, vehicles and Ohio State's renowned marching band will then make a left onto East 9th Street -- a main drag connecting the Cavs' home arena with the Rock & Roll Hall of Fame on the shores of Lake Erie.
The parade will conclude with a massive rally at Mall B, a large outdoor public space where James and other celebrities are expected to speak.
Officials have not provided an estimate for the parade's crowd, but more than 10,000 fans greeted the Cavs when they arrived at Hopkins International Airport on Monday, hours after their dramatic 93-89 win in Game 7.
The city will be well prepared, and the humongous crowd will serve as a perfect dress rehearsal for next month's Republican National Convention, which will be held at Quicken Loans.
While Cleveland hosts other parades like a popular one on St. Patrick's Day, there hasn't been a sports parade of this magnitude since 1995, when the Indians were honored for their first appearance in the World Series since 1954.
The Associated Press contributed to this report.