Precise delay control is of paramount importance in optical pump-probe measurements. Here, we report on a high-precision delay tracking technique for mechanical scanning measurements in a Mach-Zehnder interferometer configuration. The setup employs a 1.55-μm continuous-wave laser beam propagating along the interferometer arms. Sinusoidal phase modulation at 30 MHz, and demodulation of the interference signal at the fundamental frequency and its second harmonic, enables delay tracking with sampling rates of up to 10 MHz. At an interferometer arm length of 1 m, root-mean-square error values of the relative delay tracking below 10 attoseconds for both stationary and mechanically scanned (0.2 mm/s) operation are demonstrated. By averaging several scans, a precision of the delay determination better than 1 as is reached. We demonstrate this performance with a mechanical chopper periodically interrupting one of the interferometer arms, which opens the door to the combination of high-sensitivity lock-in detection with (sub-)attosecond-precision relative delay determination.