The connection between ferromagnetic (FM) fluctuations and superconductivity has been well established in FM superconductors UCoGe and URhGe. However, this connection has not been clarified in UGe2, in which pressureinduced superconductivity emerges in the FM phase. We show results focusing on the spin-echo decay rate, 1=T2, obtained using 73Ge-nuclear magnetic resonance (NMR) under pressure, which reveals longitudinal magnetic fluctuations in UGe2. These fluctuations originate in a FM critical point located at a nonzero temperature and developed near the first-order FM quantum phase transition. This remarkable feature at the high-pressure side, where a superconducting anomaly has been clearly observed, provides a connection of FM fluctuations and superconductivity in UGe2. This strongly supports a consensus that superconductivity in ferromagnets is universally mediated by anisotropic magnetic fluctuations.