The Uruguayan international is out of contract with the French Champions at the end of this season, with rumours suggesting he will leave the Parc des Princes.
Beckham is currently busy building a squad for his Miami franchise, who are due to embark on their inaugural season in the MLS from the following season.
Uruguayan journalist and former player Alvaro Izquierdo reported that Cavani – who turns 33 in February – will sign with Miami when his PSG deal expires, before joining next June.
Miami so far only have three players on their roster, after signing relatively-unknown youngsters Matias Pellegrini, Julian Carranza and Christian Makoun (all 19).
Beckham wants to add at least one marquee signing to Miami’s ranks before the 2020 season kicks off, and sees Cavani as an ideal option.
Miami’s sporting director Paul McDonough said: “It’s important for us that we found young, exciting attacking players that are on the upside of their career and have the potential to be really good players in this league.
“We will continue to build and add quality players that are the right fit for our club.
“We started with youth, but we know it’s also important to have veteran players who come from both overseas and within the league to balance out the roster over time.”
The new franchise are still yet to appoint a manager, but one ex-Premier League boss has revealed he is open to the role .
David Moyes – formerly of Man Utd – has been out of work since his sacking as West Ham boss last May.
Speaking of speculation linking him with the Miami hotseat, Moyes told talkSPORT: “It is something which looks a really interesting project.”
“So MLS is growing and Miami is a great city. It is something which I would consider as an option.”