I would say the manager still has some say based on this article from the Koeman era. Perhaps not as much as other teams, but signings in the past such as Osvaldo, Martina, Pelle and Pied have very clearly been managers picks.
http://www.telegraph.co.uk/sport/football/11366206/Gary-Neville-The-era-of-the-gaffer-is-over.html