We propose a novel, attention-based self-supervised approach to identify "claim-worthy" sentences in a fake news article, an important first step in automated fact-checking. We leverage "aboutness" of headline and content using attention mechanism for this task. The identified claims can be used for downstream task of claim verification for which we are releasing a benchmark dataset of manually selected compelling articles with veracity labels and associated evidence. This work goes beyond stylistic analysis to identifying content that influences reader belief. Experiments with three datasets show the strength of our model. Data and code available at https://github.com/architapathak/Self-Supervised-ClaimIdentification