I don't remember if this has been discussed. Did Barzini contact Tessio before Vito had died?
If so, Tessio would not only betray Michael, but also Vito, which I can't believe.
If not: There wasn't much time between Vito's death and his funeral. And Barzini must have calculated some days for Tessio to grieve for his boss and friend, before he opened up for his suggestion.