Gudjohnsen was poised to join West Ham on a one-year deal after passing a medical at Upton Park earlier this week.
The 32-year-old is available on a free transfer after leaving Stoke earlier this summer and Allardyce was keen to be reunited with the Icelandic international having worked with him during his time in charge of Bolton.
However, Greek side AEK Athens are thought to have enterted the chase for Gudjohnsen and Allardyce has decided to end his interest in the player after he hesitated in signing the deal on offer at Upton Park.
"The bottom line was that for me, Eidur had enough time to make a decision," said Allardyce.
"The time I gave him elapsed so instead of us sitting and waiting for him, I decided from my point of view that it was better to call the deal off."
Allardyce admitted he was disappointed to see the move for Gudjohnsen fall through.
"I am very disappointed because I thought the deal was done and dusted," added Allardyce. "But what happened is his father flew in last night and something happened between me finishing with him and his father arriving.
"There must have been another deal that appeared somewhere else, but irrespective of that, Eidur needed to give me a decision, he didn't, so I made the decision for both of us by nor pursing him."