Type Ia supernovae (SNe Ia) are among the most energetic events in the universe. They are excellent cosmological distance indicators due to the remarkable homogeneity of their light curves. However, the nature of the progenitors of SNe Ia is still not well understood. In the single-degenerate model, a carbon-oxygen white dwarf (CO WD) could grow its mass by accreting material from an asymptotic giant branch (AGB) star, leading to the formation of SNe Ia when the mass of the WD approaches to the Chandrasekhar-mass limit, known as the AGB donor channel. In this channel, previous studies mainly concentrate on the wind-accretion pathway for the mass-increase of the WDs. In the present work, we employed an integrated mass-transfer prescription for the semidetached WD+AGB systems, and evolved a number of WD+AGB systems for the formation of SNe Ia through the Roche-lobe overflow process or the wind-accretion process. We provided the initial and final parameter spaces of WD+AGB systems for producing SNe Ia. We also obtained the density distribution of circumstellar matter at the moment when the WD mass reaches the Chandrasekhar-mass limit. Moreover, we found that the massive WD+AGB sample AT 2019qyl can be covered by the final parameter space for producing SNe Ia, indicating that AT 2019qyl is a strong progenitor candidate of SNe Ia with AGB donors.