A private compression design problem is studied, where an encoder observes useful data Y, wishes to compress them using variable-length code, and communicates them through an unsecured channel. Since Y are correlated with the private attribute X, the encoder uses a private compression mechanism to design an encoded message C and sends it over the channel. An adversary is assumed to have access to the output of the encoder, i.e., C, and tries to estimate X. Furthermore, it is assumed that both encoder and decoder have access to a shared secret key W. In this work, the design goal is to encode message C with the minimum possible average length that satisfies certain privacy constraints. We consider two scenarios: 1. zero privacy leakage, i.e., perfect privacy (secrecy); 2. non-zero privacy leakage, i.e., non-perfect privacy constraint. Considering the perfect privacy scenario, we first study two different privacy mechanism design problems and find upper bounds on the entropy of the optimizers by solving a linear program. We use the obtained optimizers to design C. In the two cases, we strengthen the existing bounds: 1. |X|>=|Y|; 2. The realization of (X,Y) follows a specific joint distribution. In particular, considering the second case, we use two-part construction coding to achieve the upper bounds. Furthermore, in a numerical example, we study the obtained bounds and show that they can improve existing results. Finally, we strengthen the obtained bounds using the minimum entropy coupling concept and a greedy entropy-based algorithm. Considering the non-perfect privacy scenario, we find upper and lower bounds on the average length of the encoded message using different privacy metrics and study them in special cases. For achievability, we use two-part construction coding and extended versions of the functional representation lemma. Lastly, in an example, we show that the bounds can be asymptotically tight.