It emerged earlier this week that Kovac was set to depart Upton Park despite having one year remaining on his contract.
He has grown frustrated over the past few months at a lack of regular opportunities, having fallen down the midfield pecking order under Avram Grant.
With the Hammers now preparing for life in the Championship with new manager Sam Allardyce, Kovac has decided to embark on a fresh challenge.
Basel claimed top spot in Switzerland last season and have offered the former Czech Republic international a two-year deal as they prepare for Champions League football.
Kovac, 31, could face the Hammers when Basel take on Allardyce's side in a pre-season friendly on 13th July.
Kovac initially joined West Ham from Spartak Moscow in January 2009 before making the move permanent the following August.