FU Orionis systems are young stars undergoing outbursts of disc accretion and where the optical spectrum contains lines associated with both the disc photosphere and a wind component. Previous observations of the prototype FU Orionis by Herbig et al., suggested that the wind lines and the photospheric lines are modulated with periods of 14.54 and 3.54 d, respectively. We have re-observed the system at higher spectral resolution, by monitoring variations of optical line profiles over 21 nights in 2007 and have found periods of 13.48 and 3.6 d in the wind and disc components, consistent with the above: this implies variability mechanisms that are stable over at least a decade. In addition, we have found (i) that the variations in the photospheric absorption lines are confined to the blue wing of the line (centred on velocity ∼ −9 km s−1): we tentatively ascribe this to an orbiting hotspot in the disc, which is obscured by a disc warp during its receding phase. (ii) The wind period is manifested not only in blueshifted Hα absorption (as found by Herbig et al.), but also in redshifted emission of Hα and Hβ, as well as in blueshifted absorption of Na i D, Li i and Fe ii λ5018. (iii) We find that the periodic modulation of blueshifted Hα absorption (at a velocity of around −100 km s−1) is phase lagged with respect to variations in the other lines by around 1.8 d. This is consistent with a picture in which variations at the wind base first affect chromospheric emission and then low-velocity blueshifted absorption, followed – after a lag equal to the propagation time of disturbances across the wind's acceleration region – by a response in high-velocity blueshifted absorption. Such arguments constrain the size of the acceleration region to be ∼1012 cm. We discuss possible mechanisms for periodic variations within the innermost 0.1 au of the disc, including the possibility that these variations indicate the presence of an embedded hot Jupiter in this object.