Messi's current deal with Barcelona is due to expire at the end of June, and there remains a host of speculation surrounding the Argentina international's future.
Ronaldo, meanwhile, will be out of contract at Juventus next summer, and the 36-year-old has been tipped to play in North America before the end of his professional career.
Beckham has refused to rule out a move for either player as Inter Miami, now managed by Phil Neville, prepare for the start of the new MLS season in April.
"I have said from day one that our academy system is a massive part of this club and if we can bring players through here that have lived and breathed Inter Miami then that is the goal," Beckham told reporters.
"But we also know that being in Miami, our fans want to see the big stars. We already have players like Gonzalo Higuain and Blaise Matuidi who add to the glitz and glamour but also going forward we want to have the opportunity to bring some great names in.
"Leo and Cristiano have been mentioned (in the question just asked), they have been at the top of their game and been the best for the last 15 years because of hard work. There is very little luck behind it.
"The player has to be right for the team, still have ambition and want to win. So whatever player we talk to, we have to make sure that is a priority. Of course, we always want to bring in great players. Miami is a great pull for anyone and those kinds of players (Messi and Ronaldo) are who we aspire to bring here."
Inter Miami also recently completed a move for English defender Ryan Shawcross.