I think Barzini approached Tessio and Tessio "went over" when the Don(his Don)retired and left Michael in charge. Mike wasnt fighting back against Barzini's muscle, to prevent a war before he had all his plans completed,and to Tessio, and everyone else, that was a sign of weakness.Barzini probably wanted to find out if Vito really was retired and not just working with Mike as his cover before Barzini attacked the Corleones.Tessio was probably promised control of the Corleones when Mike, Clemenza and others loyal to Vito were out of the way.