Recent observation-based studies suggest that the Amazon rainforest has lost substantial resilience since 1990, indicating that the forest might undergo a critical transition in the near future due to global warming and deforestation. The idea is to use trends in a lag-1 auto-correlation of leaf density as an early-warning signal of an imminent critical threshold for rainforest dieback. Here we test whether the observed change in auto-correlations could arise from internal variability using historical and control simulations of nine sixth-generation Earth system model ensembles (Phase 6 of the Coupled Model Intercomparison Project, CMIP6). We quantify trends in the leaf area index auto-correlation from both models and satellite-observed vegetation optical depth from 1990 to 2017. Four models reproduce the observed trend with at least one historical realization whereby the observations lie at the upper limit of model variability. Three out of these four models exhibit similar behavior in control runs, suggesting that historical forcing is not necessary for simulating the observed trends. Furthermore, we do not observe a critical transition in any future runs under the strongest greenhouse gas emission scenario (SSP5-8.5) until 2100 in the four models that best reproduce the past observed trends. Hence, the currently observed trends could be caused simply by internal variability and, unless the data records are extended, have limited applicability as an early-warning signal. Our results suggest that the current rapid decline in the Amazon rainforest coverage is not foremost caused by global warming.