I don't know.
1) I've always felt that Puzo over-stretched here Vito's abilities to foresee future events.
2) Sure, contacting one of Michael's capos could be one option for Barzini. But it was very risky. What if Tessio had pretended to agree with Barzini's deal but told Michael everything? Barzini could not be sure about that.
3) Also, it makes Vito almost godlike to foresee these kind of events in detail.
4) And if they knew Barzini would approach Clemenza or Tesssio, why wouldn't Michael and Tessio simply talk to them about the situation?