This study designs a new RANS transition model by integrating an existing one-equation transition model (γ transition model) and Reynolds stress model (SSG/LRR-ω RSM). The objective of this study is twofold: 1) to propose a RANS transition model which can predict three-dimensional flow features including crossflow instabilities more accurately, compared to existing typical RANS approaches, and 2) to provide a good benchmark model by integrating reliable existing models for the future developments of RANS methods. To blend the γ transition model and SSG/LRR-ω RSM, a simple blending function is suggested here for the ω source term. Some numerical tests are conducted, including a zero-pressure-gradient flat plate, a two-dimensional single-element airfoil, a two-dimensional multi-element airfoil (30P30N), and a three-dimensional prolate-spheroid. The γ-SSG/LRR-ω RSM proposed in this study demonstrates better predictions in the numerical tests, compared to the γ-k-ω SST model and non-transition models. However, both the γ-SSG/LRR-ω RSM and γ-k-ω SST model predict different transition characteristics to the experimental results in the three-dimensional prolate-spheroid test. This might be because the transition induced by crossflow instabilities is not considered in the transition models tested in this paper, and therefore further modifications will be discussed in the presentation.