A hearing began last week before arbitrator Fredric Horowitz. After a week of sessions, three additional days are scheduled starting Oct. 15, and more days may be needed. A decision is unlikely before winter.
In his lawsuit, Rodriguez claimed MLB and Selig tried to smear the three-time AL MVP and cost him tens of millions of dollars in an attempt to cover up Selig's alleged past inaction on performance-enhancing drugs. The lawsuit claimed Selig hoped to redeem himself by going after Rodriguez.
MLB responded by calling Rodriguez's actions "desperate" and a violation of the confidentiality provisions of the league and union's Joint Drug Agreement.
On Friday, Rodriguez sued the Yankees team physician and a New York hospital in New York Supreme Court in the Bronx, claiming they mishandled his medical care during the 2012 American League playoffs by letting him play after failing to inform him that a test revealed an injury to his left hip joint.
The lawsuit said Rodriguez then further injured himself, forcing him to suffer hospitalization, disability and emotional distress.
During the 2012 postseason, Rodriguez had three hits from 25 at bats with no RBIs before undergoing left hip surgery in January that prevented him from rejoining the Yankees until August.
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AP Sports Writer Ronald Blum contributed to this report