An apparatus for controlling the injection pressure of an aqueous polymer solution in a subsea oil wellhead, includes a linear pressure reducer in the form of a tube connected in series to the main injection line that is capable of absorbing the majority of the pressure drop and a choke that is capable of being regulated in order to enable a pressure control between 0 and 10 bar. Also, a process of offshore enhanced oil recovery by injection of an aqueous polymer solution uses the apparatus.