Turnbull is available on a free transfer after being released by Chelsea earlier this summer.
The 28-year-old found himself playing back-up to Petr Cech during his four-year spell at Stamford Bridge, making just 18 appearances for Chelsea.
RKC Waalwijk are looking to bring in a new goalkeeper as Jeroen Zoet has returned to PSV Eindhoven after spending two seasons on loan at RKC.
Turnbull featured for RKC in Saturday's pre-season defeat to Rot Weiss Oberhausen in Germany.
RKC coach Erwin Koeman will now weigh up now whether or not to pursue his interest in former Middlesbrough keeper Turnbull.