The 22-year-old, who scored an impressive 15 goals for the Danish Superliga side last season, was linked with a late move to Selhurst Park.
However, the proposed loan deal collapsed hours before the close of the summer transfer window after Brondby decided to reject the offer.
"We did receive an offer from a big English club at around 10pm on deadline day," Rud confirmed to the club's official website.
"However, after a short time of consideration we decided to reject the offer because we did not want to send him out on loan shortly before the transfer window closed."
The 6ft 7in Makienok turned down a move to Russian outfit Terek Grozny earlier in August, but he is now expected to consider his options in January.