The 22-year-old centre-back is set to cost Dortmund around €21.5m ($26.2m) from Swiss side Basel, according to German daily 'Bild' and football magazine 'Kicker'.
Akanji, who has made four international appearances, was reportedly also being chased by Premier League giants Man United and Liverpool.
Dortmund had reportedly initially offered €15m for Akanji, who cost Basel €700,000 from Winterthur just two years ago, but upped their bid after problems in defence so far this season.
Dortmund have yet to confirm Akanji's transfer ahead of their first Bundesliga match of 2018 at home to Wolfsburg on Sunday.