The Norwegian outfit are thought to have rejected an offer of around £2.1 million for the versatile Sierra Leone international, and are holding out for closer to £4.2m despite just one year remaining on his current contract.
Bangura has urged his employers to allow him to move to Selhurst Park, telling the Haugesund Avis: "I am very disappointed."
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"I only have a few months left on my contract before I can go for free, so the choice should not be that difficult."
The 25-year-old was previously on the books of London rivals Watford after impressing at the Under-17 World Cup in 2003, but was loaned to Honefoss after failing to attain a work permit.
He remained at the Norwegian second division club after his contract at Vicarage Road expired, and his excellent form earned him a move to top-flight outfit Haugesund in 2010.
With Sierra Leone currently ranked inside Fifa’s top-70 nations, he would now be eligible for a work permit should Palace manage to strike a deal.