We assess the proximity of a system to a bifurcation point using a degenerate fingerprinting method that estimates the declining decay rate of fluctuations in a time series as an indicator of approaching a critical state. The method is modified by employing Detrended Fluctuation Analysis (DFA) which improves the estimation of short-term decay, especially in climate records which generally possess power-law correlations. When the modified method is applied to GENIE-1 model output that simulates collapse of the Atlantic thermohaline circulation, the bifurcation point is correctly anticipated. In Greenland ice core paleotemperature data, for which the conventional degenerate fingerprinting is not applicable due to the short length of the series, the modified method detects the transition from glacial to interglacial conditions. The technique could in principle be used to anticipate future bifurcations in the climate system, but this will require high-resolution time series of the relevant data.