Roma great De Rossi left the Serie A club at the end of the 2018-2019 season but insisted he was not planning to retire.
He is consequently a free agent and Burdisso, a former team-mate of De Rossi's at the Stadio Olimpico, is keen to see the midfielder in Boca colours.
"A lot has been said about De Rossi," the former Argentina international told Closs Continental. "I know him and he opened the doors to coming here.
"He would be the cherry on top of the cake. I think Daniele could play for Boca. I know him as I was with him for five years.
"He values the team highly and watches all our games. Here there are things that China and MLS do not have, such as the passion of the people."
However, ESPN reports runaway MLS leader LAFC is set to complete a deal for World Cup winner De Rossi.
The American club sold Andre Horta to Braga this week, freeing a Designated Player slot for a high-profile player such as the 35-year-old.
De Rossi would join existing DPs Carlos Vela and Diego Rossi, who have fired LAFC six points clear with a game in hand in the Supporters' Shield race.