Busquets, 34, is out of contract with Barca at the end of June and, after announcing he will not extend his deal, has been considering offers from around the world.
However, following a series of positive conversations with the hierarchy at Inter Miami, who are part-owned by David Beckham, a move to South Florida and a reunion with Messi is edging closer.
Sources say a deal is still not 100% done, with the two Saudi clubs reluctant to give up on Busquets, but a final decision is expected soon.
Busquets' camp expect to field the final offers from Al Nassr and Al Hilal on Friday as they try to convince the former Spain international to join the influx of players moving to the Middle East.
ESPN first revealed in January that Al Nassr, who signed Cristiano Ronaldo last year, had made a move to sign Busquets as a free agent. The initial offer six months ago was worth around €13 million ($14.1m) net but sources have told ESPN that figure has "significantly increased" since then.
Sources added that Busquets will earn much less in Miami than he could in Saudi Arabia, but his family have always been more keen on a move to the United States. The player has previously spoken of his desire to end his career in the U.S. too.
Playing alongside Messi again -- the two spent 13 years as teammates in Barca's first team -- is also a draw, although sources say Messi has remained in the background and not been involved in Busquets' decision or Miami's push for the midfielder.
The two remain close friends off the pitch and dined together just a few weeks ago in Barcelona along with Spain left-back Jordi Alba, who is also drawing interest from Miami following the announcement he will leave Barca this summer.