Cardiff will allow long-serving Kevin McNaughton to leave this summer along with Danny Gabbidon and Nicky Maynard.
Highly-rated Atletico Madrid midfielder Saul Niguez has signed a new five-year contract.
Bangura recently returned from a loan spell at former club AIK last month but failed to force his way into the starting line-up, despite scoring in a friendly against Steaua Bucharest this month.
The Sierra Leone forward has failed to score in 13 competitive appearances since joining Celtic for about £2million in August 2011.
The 23-year-old told Elfsborg's official website that he first heard of a possible transfer on Wednesday, but that both clubs had previously been in contact.