Strahan was one of seven players elected to the Pro Football Hall of Fame on Saturday, four days after Sapp was critical of the defensive end.
Sapp, a 2013 Hall of Fame inductee who currently works as an analyst for NFL Network, said he approached Strahan on Saturday night to apologize.
"I hugged him and wouldn't let him go," Sapp said Sunday on NFL Network. "I said, 'Please forgive me.' I said, 'Dumb Lex Luthor came to the city of Gotham and pulled Superman's cape. How foolish could I have been, my man? Please forgive me.'
"If he tells the story any different, he's lying to you, America, because I sure did apologize. I'm going to beg for forgiveness because there's a party in Canton, and I promise you, I'm not going to miss it."
Sapp did not disclose whether Strahan responded to his apology.
Strahan recorded 141.5 sacks, the fifth-highest total in NFL history, in his 15-year career and set the league's single-season record with 22.5 sacks in 2001.
"I don't think his résumé stacks up," Sapp said. "Four straight Pro Bowls and a mythical sack record. When you really measure it up, he comes up short, except you all are giving it to him."
Strahan fired back at Sapp this past week, saying that the "tiger does not pay attention to the opinion of the sheep."
ESPN.com Giants reporter Dan Graziano and The Associated Press contributed to this report.