The normal state of the unconventional superconductor Sr₂RuO₄ (T_c = 1.45 K) is a textbook Fermi Liquid (FL), with signatures of long-lived quasiparticles seen in resistivity, optical conductivity and quantum oscillations below T_FL∼30 K. However at higher temperatures, its resistivity increases unconstrained and eventually exceeds the Ioffe-Regel limit, indicative of strange metallic behavior. We present evidence from two experiments for the presence of nematic fluctuations (along [100]) in Sr₂RuO₄ above T_FL. From ultrasound experiments, we find a large (∼20%) anomalous softening in the B_1g shear modulus (c₁₁ - c₁₂)/2 between 300 K and 35 K, below which its behavior is standard. In contrast, the B_2g shear modulus c₆₆ shows standard behavior over the entire range. Motivated by this evidence of [100] nematicity, and to rule out a lattice origin of the effect, we then performed elastoresistivity measurements. We measure a diverging nematic susceptibility in the B_1g channel which saturates around 60 K, about the same temperature where (c₁₁ - c₁₂)/2 starts to show deviations from Curie-Weiss behavior. Our study suggests that nematic correlations might be the reason behind non-FL physics in Sr₂RuO₄, and hints towards Sr₂RuO₄ being close to a nematic quantum critical point.