Background: Functional mobility and muscle strength are well known risk factors for sarcopenia. Furthermore, possible associations have been suggested between predisposing factors of sarcopenia and reaction time among the elderly. This study aims to analyze possible associations of functional mobility and muscle strength and reaction times in a population of people aged >60 years. Methods: A total of 290 older people (69.35 ± 5.55 years) participated in this study. The following parameters were assessed: optoacoustic lower-limb reaction time (OALLRT); acoustic lower-limb reaction time (ALLRT); optic lower-limb reaction time (OLLRT, using an optical detection system), functional mobility (through the timed up-and-go test) and muscle strength (using a dynamometer). Results: Our results show that lower values of muscle strength were associated with increased reaction times in OALLRT (β = −0.170; 95% confidence interval −0.011–0.000; R2 = 0.237; p = 0.035) and in ALLRT (β = −0.228; 95% confidence interval −0.011–0.002; R2 = 0.199; p = 0.006). Conclusion: Increased muscle strength (which at low values are risk factors for sarcopenia) was associated with decreased reaction times in people >60 years of age.