Beckham, who is a majority shareholder at the MLS club, is keen on building a high profile project in the US, including some of the biggest names in European football.
Beckham claimed in an interview with Mundo Deportivo that he has the ambition to bring the best players to his new project, with salaries unlikely to be a barrier.
“I know that in Miami our fans want to see big stars,” he said.
“We already have players like Gonzalo Higuaín and Blaise Matuidi, but in the future we want to have the opportunity to bring in big names.
“Leo and Cristiano have been mentioned, they have been at the top of their game and they have been the best for the last 15 years.
“We want to attract the best players. Miami has a great power of attraction and these are the players we want to bring here.”
Messi has consistently stated he will wait until the end of the season before making a decision on whether or not to stay at the Camp Nou in 2021.
Premier League giants Manchester City are also likely to reignite their long standing interest in him, alongside Ligue 1 champions Paris Saint-Germain.