In this paper, we generalize the achievability of variable-length coding from two viewpoints. One is the definition of an overflow probability, and the other is the definition of an achievability. We define the overflow probability as the probability of codeword length, not per symbol, is larger than ηn and we introduce the ε-achievability of variable-length codes that implies an existence of a code for the source under the condition that the overflow probability is smaller than or equal to ε. Then we show that the ε-achievability of variable-length codes is essentially equivalent to the ε-achievability of fixed-length codes for general sources. Moreover we show the condition of ε-achievability for some restricted sources given ε.