A computational framework of time integrators, termed GS4-1, is described for use in first-order transient heat transfer problems. The framework consists of a family of algorithms by design with desirable features including second-order accuracy, unconditional stability, and zero-order overshoot. Of noteworthy mention and significance is the fact that the present computational framework naturally inherits features that permit selective control of the high-frequency damping for the primary variable and its time derivative, respectively, which is a new feature not available in existing methods. Consequently, the developments enable physically representative and accurate solutions of both variables. We first present, in Part 1, the derivation of such a framework with application to linear heat transfer problems. The significance of the present developments in capturing the problem physics via the selective control feature is demonstrated through an application to a simple illustrative one-dimensional problem for which an analytic solution is available. The extension of the framework for nonlinear cases is presented in Part 2.