Under European Association of Zoos and Aquaria (EAZA) rules, in-breeding between giraffes is to be avoided.
Although Marius is healthy, his genes are already well represented at the zoo and he cannot be taken in by the 300 other EAZA-affiliated zoos.
Castration is considered cruel with "undesirable effects." while releasing him in the wild is thought unlikely to succeed.
The giraffe's impending death sparked outrage online, with more than 5,000 people signing a "Save Marius" Facebook petition. More than 3,000 people signed a similar Danish-language online petition and nearly 24,000 an English-language version.
There were reportedly several attempts made to save Marius.
A Swedish zoo, which is not part of the EAZA network, tried in vain to get Marius transferred, the Expressen daily reported.
And another daily, Denmark's Ekstrabladet, quoted a Danish promoter living in Los Angeles, Claus Hjelmbak, as saying he had found a buyer for the animal.
"One of my close friends, a billionaire, said that he wanted to transfer a few million so we could save the giraffe," Hjelmbak was quoted as saying.