Mechanical rock properties are often determined using sonic log data—compressional velocity (VP) and shear velocity (VS). However, a sonic well log is not always acquired due to deteriorated hole condition (i.e., hole washout), sonic tool failures, especially in high-pressure, high-temperature (HPHT) wells, and relatively high cost. This paper introduces two data-driven models, namely artificial neural network (ANN) and random forest (RF), to estimate VP and VS across different formations that are characterized by deep burial depth and strong heterogeneity. Two types of actual field data were used to develop the models: (i) drilling surface parameters, which include flow rate, standpipe pressure, rotary speed, and surface torque, and (ii) acoustic velocities VP and VS, which were acquired by a conventional sonic log. Well-1 and Well-2 with data points of 6,846 were used to develop the models, while Well-3 with 1,016 data points was used to evaluate the capability of the developed models to generalize on an unseen data set with different statistical behavior. Furthermore, Well-3 was used to compare the accuracy of the developed models with the earliest published correlations in estimating the VS. The results showed that the RF outperformed the optimized ANN in estimating VP and VS in Well-3. The RF predicted the VP with a low average absolute percentage error (AAPE) of 0.9% and correlation of coefficient (R) of 0.87, while the AAPE and R were 6.7 % and 0.45 in the case of ANN. Similarly, the RF estimated the VS with an AAPE of 1.1% and R of 0.85, whereas the ANN predicted the VS with an AAPE of 9.5% and R of 0.40. Furthermore, the RF was the most accurate in determining VS in Well-3 compared to the earliest published correlations.