I'm with Oli: Tessio approached Barzini. In the fishtank scene, it's Tessio who's complaining about Barzini. And, when Vito asks, "Do you trust my judgment?", Clem answers fervently, "Always, Godfather." Tessio hisses, "Yes-s-s-s-..." When leaving, Tess gives the left-hand handshake--a sure sign of perfidy-to-come.

Since Tessio brought Barzini's request for a meeting at Vito's burial, it seems likely that no more than a few weeks elapsed until the massacre.