Context. Although Type Ia supernovae (SNe Ia) are important in many astrophysical fields, the nature of their progenitors is still unclear. A new version of the single-degenerate model has been developed recently, the common-envelope wind (CEW) model, in which the binary is enshrouded in a common envelope (CE) during the main accretion phase. This model is still in development and has a number of open issues, for example what is the exact appearance of such a system during the CE phase? Aims. In this paper we investigate this question for a system with a massive CE. Methods. We use a thermally pulsing asymptotic giant branch (TPAGB) star with a CO core of 0.976 M-circle dot and an envelope of 0.6 M-circle dot to represent the binary system. The effects of the companion's gravity and the rotation of the CE are mimicked by modifying the gravitational constant. The energy input from the friction between the binary and the CE is taken into account by an extra heating source. Results. For a thick envelope, the modified TPAGB star looks similar to a canonical TPAGB star but with a smaller radius, a higher effective temperature, and a higher surface luminosity. This is primarily caused by the effect of the companion's gravity, which is the dominant factor in changing the envelope structure. The mixing length at the position of the companion can be larger than the local radius, implying a breakdown of mixing-length theory and suggesting the need for more turbulence in this region. The modified TPAGB star is more stable than the canonical TPAGB star and the CE density around the companion is significantly higher than that assumed in the original CEW model. Conclusions. Future work will require the modelling of systems with lower envelope masses and the inclusion of hydrodynamical effects during the CE phase.