We construct the Frobenius structure on a rigid connection Be_{\hat G} on G_m for a split reductive group \hat G introduced by Frenkel-Gross. These data form a \hat G-valued overconvergent F-isocrystal Be^†_{\hat G} on G_{m,F_p}, which is the p-adic companion of the Kloosterman \hat G-local system Kl_{\hat G} constructed by Heinloth-Ngô-Yun. By studying the structure of the underlying differential equation, we calculate the monodromy group of Be^†_{\hat G} when \hat G is almost simple (which recovers the calculation of monodromy group of Kl_{\hat G} due to Katz and Heinloth–Ngô–Yun), and prove a conjecture of Heinloth-Ngô-Yun on the functoriality between different Kloosterman \hat G-local systems. We show that the Frobenius Newton polygons of Kl_{\hat G} are generically ordinary for every \hat G and are everywhere ordinary on |G_{m,F_p}| when \hat G is classical or G_2.