In our previous study (Partanen, J. I.; Partanen, L. J.; Vahteristo, K. P., J. Chem. Eng. Data 2019, 64, 16-33), we presented a fully traceable two-parameter Hückel equation with parameters B and b1 for activity and osmotic coefficients in dilute KCl(aq) in the temperature range of (273.15 to 383.15) K. This equation applies within experimental error to almost all thermodynamic data in the literature at least up to a molality of 0.2 mol·kg-1. In our Hückel model, parameter B is treated as a constant, whereas parameter b1 depends quadratically on the temperature. In the present study, the same model is applied to the molar enthalpies of the components in KCl solutions. No new parameter estimations are necessary for the treatment of these calorimetric data. We also extend now the previous enthalpy results for dilute NaCl solutions (see Partanen, J. I.; Partanen, L. J.; Vahteristo, K. P., J. Chem. Eng. Data 2017, 62, 2717-2632) up to 373 K because in that study the treatment of calorimetric data was limited only up to 353 K. In a future publication (Part 2 of this study), it will be shown that this Hückel model applies additionally well to the heat-capacity literature available for dilute KCl solutions. Here, we also consider a second parametrization of the Hückel equation for KCl solutions obtained in a previous work (Partanen, J. I., J. Chem. Eng. Data 2016, 59, 286-306). This is seen to apply better to enthalpy data in less dilute KCl solutions up to the saturated solutions in the range of (298 to 303) K. However, the alternative parametrization is not fully traceable and not as accurate as our primary one. Following the success of these Hückel models, we supplement the thermodynamic tables for KCl solutions with the relative apparent and partial molar enthalpies of KCl in these solutions from (273.15 to 373.15) K. We also give here the values of these quantities for NaCl in dilute aqueous NaCl solutions at 373.15 K. We have good reason to believe that the new tables contain the most reliable values available for the enthalpy quantities of dilute NaCl and KCl solutions.