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Natural gas network is important for residential heating, industrial manufacturing, and electricity generation. Although it is reliable and resilient to local disruptions, extreme situations such as natural disasters and political conflicts can degenerate its capability of gas transportation and delivery, influencing other social activities. Evaluating loadability regions of natural gas networks is hard due to nonlinear constraints. This paper proposes a fast computational tool for feasibility screening of natural gas load profiles. It first establishes the theoretical results on the convexity of loadability regions with sufficient conditions. Then, an asymptotic algorithm is applied to compute a sequence of inner polytopes that converges to the convex loadability region. Each polytope in the sequence can serve as a certificate for feasibility. The conservativeness of this inner estimation will decline along the monotone sequence. The algorithm is testified on a modified realistic Belgian natural gas system with multi-dimensional load profiles.