Beasley, an unrestricted free agent, flew to Phoenix to meet with the Suns on Wednesday. The Suns flew to Los Angeles to meet Beasley earlier this week.
Sources say the Suns are likely to make the talented small forward a contract offer before he leaves town.
While Beasley has been dogged by bouts of immaturity throughout his four-year career, the Suns are not scared off by his reputation and instead are fond of him as a player and a person, according to sources.
Suns general manager Lance Blanks has a history of working with troubled players dating back to his time as an executive with the Cleveland Cavaliers. Blanks was instrumental in bringing volatile guard Delonte West to the Cavs and continues to have a strong relationship with him.
With star point guard Steve Nash on the verge of joining another club, the Suns have emerged as one of the more active teams in free agency.
In addition to courting Beasley and working on potential sign-and-trade deals with New York and the Los Angeles Lakers for Nash, the Suns offered New Orleans shooting guard Eric Gordon a maximum $58 million over four years.
While some sources say New Orleans will match the offer for Gordon, others say Gordon's public statements about preferring Phoenix to New Orleans could potentially lead to discussions on a sign-and-trade deal between the clubs.