• Have a diagnosis of CLL requiring treatment according to the International Workshop on CLL (iwCLL) criteria (Hallek et al 2018)

• Eastern Cooperative Oncology Group (ECOG) performance score (PS) of ≤ 2

• Adequate bone marrow (BM) function independent of growth factor or transfusion support, within 2 weeks of screening, at screening as defined by the protocol unless cytopenia is clearly due to marrow involvement of CLL

• Adequate liver function unless directly attributable to the participant's CLL

• Life expectancy \> 6 months

• For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods that result in a failure rate of \< 1% per year, and agreement to refrain from donating eggs during the treatment period and for at least 3 months after the last dose of mosunetuzumab and 3 months after the last dose of tocilizumab (if applicable)

• For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use a condom, and agreement to refrain from donating sperm as defined by the protocol

• Participants must have been taking a BTKi for at least 12 months, have demonstrated evidence of progressive disease while receiving the BTKi and require additional salvage therapy as assessed by their treating physician. Participants should be able to continue their previously prescribed BTKi at a stable dose throughout the study screening period and for the first two cycles of mosunetuzumab administration

• Non-US participants only