Most existing methods for automatic fact-checking start with a precompiled list of claims to verify. We investigate the understudied problem of determining what statements in news articles are worthy to fact-check. We annotate the argument structure of 95 news articles in the climate change domain that are fact-checked by climate scientists at climatefeedback.org. We release the first multi-layer annotated corpus for both argumentative discourse structure (argument types and relations) and for fact-checked statements in news articles. We discuss the connection between argument structure and check-worthy statements and develop several baseline models for detecting check-worthy statements in the climate change domain. Our preliminary results show that using information about argumentative discourse structure shows slight but statistically significant improvement over a baseline of local discourse structure.