The ex-quarterback personally gave the advice to Burrow's family Sunday night during a Davey O'Brien Award event honoring the LSU star.
"They're Ohio guys," Bartkowski, the Atlanta Falcons' top pick in 1975, said, according to Drew Davison of the Fort Worth Star-Telegram. "I might've offended them by telling them that, if it's the Bengals, I think I'd pull an Eli Manning on that one. I said, you've got a chance to do that. That's happened. (John) Elway kind of set the tone, then the Mannings delved into it after Eli was picked by San Diego."
Manning famously refused to play for the then-San Diego Chargers, forcing them to trade his rights to the New York Giants during the 2004 draft.
Elway, meanwhile, threatened to pursue a baseball career after being taken No. 1 in 1983 by the then-Baltimore Colts, who eventually traded the Hall of Famer to the Denver Broncos.
Burrow is the favorite to be the first player taken after a historic senior season, in which he led the Tigers to the national championship and won the Heisman Trophy.
However, Burrow's comment Sunday that he has "leverage" as the presumptive top overall pick helped further fuel the speculation that he would prefer to play elsewhere.
"(The Bengals) have their process and I have my process," said Burrow. "We haven't even gotten to the (NFL) combine yet. There's a lot of things that happen leading up to the draft and a lot of information gathered."
Burrow's family has pushed back against the idea that the quarterback wouldn't be happy if selected by Cincinnati.
The Bengals have never won a Super Bowl and their last playoff win was in 1991.