The leakage of the injection–production string is one of the important hidden dangers for the safe and efficient operation of underground salt cavern gas storage. Although distributed fiber optic temperature measurement system (DTS) can accurately locate the position of the string leakage port, how to establish the quantitative relationship between the temperature difference and leakage rate of the leakage port still needs further exploration. This paper proposes a new quantitative prediction model based on a DTS for the leakage monitoring of the injection–production string of salt cavern gas storage. The model takes into account the gas’s physical parameters, unstable temperature conditions, and the Joule–Thomson effect. In order to verify the accuracy of the model, a simulation experiment of string leakage based on a DTS was carried out. The test results show that the relative deviation between the predicted leakage rate and the measured value is less than 5% compared with the calculated value. When the leakage rate drops to 0.16 m³/h and the temperature range is less than 0.5 °C, it is difficult to accurately monitor the DTS. The results of this study help to improve the early warning time of underground salt cavern gas storage string leakage.