Measurement-constrained datasets, often encountered in semi-supervised learning, arise when data labeling is costly, time-intensive, or hindered by confidentiality or ethical concerns, resulting in a scarcity of labeled data. In certain cases, surrogate variables are accessible across the entire dataset and can serve as approximations to the true response variable; however, these surrogates often contain measurement errors and thus cannot be directly used for accurate prediction. We propose an optimal sampling strategy that effectively harnesses the available information from surrogate variables. This approach provides consistent estimators under the assumption of a generalized linear model, achieving theoretically lower asymptotic variance than existing optimal sampling algorithms that do not use surrogate data information. By employing the A-optimality criterion from optimal experimental design, our strategy maximizes statistical efficiency. Numerical studies demonstrate that our approach surpasses existing optimal sampling methods, exhibiting reduced empirical mean squared error and enhanced robustness in algorithmic performance. These findings highlight the practical advantages of our strategy in scenarios where measurement constraints exist and surrogates are available.