Ronny Deila stressed his desire to keep Fraser Forster at Celtic following the 4-0 victory over KR Reykjavik.
Executive vice-chairman Ed Woodward insists Manchester United are ready and willing to break the British transfer record.
Sky Sports revealed earlier on Tuesday that the Black Cats had made a move for the American forward as they continue to reshape their squad under new manager Paolo Di Canio.
But their opening bid has been knocked back by AZ, who have already banked 8 million euros after allowing Adam Maher to join PSV Eindhoven and are not under pressure to sell.
An AZ spokesman told Dutch press agency ANP: "A first bid was insufficient. We have let them know that. Now we have to wait."
Altidore, who has already had a spell in England with Hull City in 2009, enjoyed the best season of his career last term after scoring 23 league goals for AZ Alkmaar.
Sunderland must now consider whether to return with an improved offer for the 23-year-old, or turn their attention to alternative targets.