West Brom have sacked Nicolas Anelka citing "gross misconduct" after the striker claimed on Twitter he had quit the club.
Mauricio Pochettino expects to be at Southampton next season and is looking forward to working with chairman Ralph Krueger.
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.