Information-theoretic security is considered in the paradigm of network coding in the presence of wiretappers, who can access one arbitrary edge subset up to a certain size, also referred to as the security level. Secure network coding is applied to prevent the leakage of the source information to the wiretappers. In this two-part paper, we consider the problem of secure network coding when the information rate and the security level can change over time. In the current paper (i.e., Part I of the two-part paper), we focus on the problem for a fixed security level and a flexible rate. To efficiently solve this problem, we put forward local-encoding-preserving secure network coding, where a family of secure linear network codes (SLNCs) is called local-encoding-preserving if all the SLNCs in this family share a common local encoding kernel at each intermediate node in the network. We present an efficient approach for constructing upon an SLNC that exists a local-encoding-preserving SLNC with the same security level and the rate reduced by one. By applying this approach repeatedly, we can obtain a family of local-encoding-preserving SLNCs with a fixed security level and multiple rates. We also develop a polynomial-time algorithm for efficient implementation of this approach. Furthermore, it is proved that the proposed approach incurs no penalty on the required field size for the existence of SLNCs in terms of the best known lower bound by Guang and Yeung. The result in this paper will be used as a building block for efficiently constructing a family of local-encoding-preserving SLNCs for all possible pairs of rate and security level, which will be discussed in the companion paper (i.e., Part II of the two-part paper).