You can try b4, and if not, then c3 or Nc3.
Italian Game 4.0-0 d6!?
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4...d6 5.c3 does look like the most logical response. After 5...Nf6 though White should play 6.Re1. If Black castles White can NOW play d4, meet ..Bb6 with h3, and White's center holds up. If Black plays 6...Bg4 instead White should just play h3 (kicking it back to h5), d3, Bb3, Nbd2-f1-g3, all standard Guioco stuff. White may not have any objective advantage but is probably banking on just getting to a familiar middlegame, anyways.
Side note: What do you think is wrong with Black's position after 4...Nf6 5.d4 Bxd4 6.Nxd4 Nxd4 7.f4 d6 8.fxe5 dxe5 9.Bg5 Be6 ?
Is there a more ambitious line against this move?