We construct a network from climate records of temperature and geopotential-height in two pressure levels at different geographical sites in the North Atlantic. A link between two sites represents the cross-correlations between the records of each site. We find that within the different phases of the North Atlantic Oscillation (NAO) the correlation values of the links in the climate network are significantly different. By setting an optimized threshold on the correlation values, we find that the number of strong links in the network increases during times of positive NAO indices, and decreases during times of negative NAO indices. We find a pronounced sensitivity of the network structure to the NAO oscillations which is significantly higher compared to the observed response of spatial average of the climate records. Our results suggest a new measure that tracks the NAO pattern.