Conductance and its bias dependence have been measured on Au and Ag breaking nanocontacts for biases up to 3.0 V at room temperature in ultrahigh vacuum. Under high-bias/high-current conditions, both Au and Ag contacts often show a characteristic conductance fluctuation when the conductance attains a certain critical value G_th. This critical conductance ranges from about 10G₀ to about 50G₀ (G₀ ≡ 2e²/h is the quantum unit of conductance) and increases with the bias. When G_th is plotted against the contact current I, we obtained a linear I-G_th plot for Au contacts. Since G_th is in a semiclassical regime and hence should be proportional to the cross sectional area of the contact, the slope of the I-G_th plot represents a critical current density for the onset of the conductance fluctuation. These observations indicate that the conductance fluctuation is due to certain current-induced contact instability. When the current density exceeds the critical value, a contact becomes unstable and tends to be ruptured, leaving small chances of further necking deformation down to a single-atom contact. [DOI: 10.1380/ejssnt.2004.125]