City are nearing an agreement of around €17.5m for the defender, who has a year left on his contract at Dortmund.
The 27-year-old is set to leave the Bundesliga side after four-and-a-half years during which he made more than 150 appearances.
He has also been capped 41 times by Switzerland. Akanji could come up against Dortmund for City this season after they were drawn together in Group G of the Champions League.
Speaking at a news conference on Tuesday, manager Pep Guardiola left the door open for more business to be done before the close of the transfer window, insisting that he "didn't know" if there would be ins and outs ahead of Thursday's deadline.
City have started the season well but have conceded five goals in their last two games and have gone 2-0 down in four of their last six league matches stretching back to last season.
"In six years in the Premier League we concede fewer chances," Guardiola said on Tuesday ahead of Nottingham Forest's visit to the Etihad Stadium.
"In the last game against Crystal Palace, we concede few; one corner, one free kick and one other shot.
"Always we score a lot of goals and concede few and now we are scoring a lot but conceding a lot and if we don't stop that it will be difficult.
"We cannot always score four or five goals to win the games. It is the transitions and the set-pieces and if we do not have more control we will suffer."