Context. It is well known that the fast solar wind originates from coronal holes (CHs). However, the question whether it can also originate from quiet Sun regions has not yet been answered. Aims.To study this problem we analyze SOHO data obtained from observations made in a quiet Sun area. The data set includes far-ultraviolet data from SUMER, magnetic field data from MDI, and extreme-ultraviolet data from EIT. Methods. We make a potential-field extrapolation of the coronal magnetic field and calculate the field lines from the photosphere up to 80 Mm height. Those field lines which can be traced from the bottom to the top of the extrapolation box are called (locally) open field lines. By a combined analysis of the coronal magnetic field structures inferred from MDI data, the flows indicated by the Ne viii Doppler shifts in the SUMER data, and the Fe xii radiance images from EIT, it is possible to study this problem in depth. Results. We find that most of the sites with plasma outflow, which can be recognized by the Ne viii blue shift, are not located in regions with an open magnetic field. Most likely, these outflows just correspond to plasma being delivered to magnetic loops. It is further found that, in a cross-section plane located at a height of 25 Mm, the pattern of open field lines intersecting that plane is consistent with the dark pattern of low radiance in the image of the Fe xii 19.5 nm line. Usually, small dark regions are considered to represent small CHs, and thus are assumed to be sources of the solar wind. However, since here the source of the low emission appears to be located at a height of only 25 Mm, it seems more likely that this radiation originates near the foot points of large coronal loops. Conclusions. Previous results obtained at middle latitudes on the quiet Sun indicated that sizable outflow velocities occur at the intersections of the network boundaries. This finding is also confirmed here. However, we could not identify most of these intersections as sources of the solar wind. Only a few small outflow regions might be sources. Yet, one dark area that we found on the EIT map seems to be connected with open field lines, and therefore it could be a source of the solar wind.