Binary stellar evolution has been studied as important pathway to initiate various transient events like supernovae (SNe). Although the common envelope (CE) in a binary, outcomes of the CE and conditions for the SN explosion during the CE phase are uncertain, it has been suggested that SN explosions can be triggered during the CE phase. In this work, we explore formation and evolution routes of carbon/oxygen (CO) WDs binaries in order to investigate mergers of CO WDs and cores of non-degenerate stars during the CE phase as possible origins for SNe by considering several binary physical models. Evolution of CO WD + intermediate mass normal star binaries lead to mergers during the CE phases still may trigger type Ia SNe (SNe Ia) interacting with circumstellar material under different models. Mergers between CO WDs and cores of He stars within the CE phases are rare comparing to other CE merger events. Above two channels may produce peculiar SNe Ia such as over-luminous/super-chandrasekhar mass SNe Ia as certain fraction of them have combined core masses ≥ 2M . In the channel of CO WD + massive star (≥ 8M ) CE events, we find that rates of mergers between CO WDs and He cores of massive stars are (0.85 -12.18)×10 −5 M −1 which may initiate type II superluminous SNe like SN 2006gy, and delay times from this scenario are in the range of 34 -120 Myr. Our results based on the core merger detonation model are comparable with observational results of peculiar SNe.