We use visual information to guide our grasping movements. When grasping an object with a precision grip, the two digits need to reach two different positions more or less simultaneously, but the eyes can only be directed to one position at a time. Several studies that have examined eye movements in grasping have found that people tend to direct their gaze near where their index finger will contact the object. Here we aimed at better understanding why people do so by asking participants to lift an object off a horizontal surface. They were to grasp the object with a precision grip while movements of their hand, eye and head were recorded. We confirmed that people tend to look closer to positions that a digit needs to reach more accurately. Moreover, we show that where they look as they reach for the object depends on where they were looking before, presumably because they try to minimize the time during which the eyes are moving so fast that no new visual information is acquired. Most importantly, we confirmed that people have a bias to direct gaze towards the index finger’s contact point rather than towards that of the thumb. In our study, this cannot be explained by the index finger contacting the object before the thumb. Instead, it appears to be because the index finger moves to a position that is hidden behind the object that is grasped, probably making this the place at which one is most likely to encounter unexpected problems that would benefit from visual guidance. However, this cannot explain the bias that was found in previous studies, where neither contact point was hidden, so it cannot be the only explanation for the bias.