Inclusion Criteria:  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  Exclusion Criteria:  Pregnant or breastfeeding, or intending to become pregnant during the study or within 3 months after the final dose of mosunetuzumab and tocilizumab (if applicable) Participants who have received any of the following treatments prior to study entry: treatment with mosunetuzumab or other CD20/CD3-directed bispecific antibodies; allogenic stem cell transplant Participants who have received any of the following treatments, whether investigational or approved, within the respective time periods prior to initiation of study treatment: radiotherapy within 2 weeks prior to the first dose of study treatment; autologous stem cell transplant within 100 days prior to first study treatment; CAR T-cell therapy within 30 days before first study treatment; use of monoclonal antibodies or antibody-drug conjugates within 4 weeks prior to first study treatment; systemic immunosuppressive medications (including but not limited to cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor agents) within 2 weeks prior to the first dose of study treatment; any other anti-cancer therapy, whether investigational or approved, including but not limited to chemotherapy within 4 weeks or 5 half-lives of the drug, whichever is shorter, prior to initiation of study treatment; other prior cancer immunotherapy not explicitly defined by the protocol is to be discussed with the medical monitor to determine eligibility Received a live, attenuated vaccine within 4 weeks before the first dose of study treatment, or in whom it is anticipated that such a vaccine will be required during the study period or within 5 months after the final dose of study treatment Transformation of CLL to aggressive non-Hodgkin's lymphoma (NHL) History of severe allergic or anaphylactic reactions to humanized or murine monoclonal antibody therapy (or recombinant antibody-related fusion proteins) Contraindication to tocilizumab History of prior malignancy except for conditions defined by the protocol Participants with infections requiring intravenous (IV) treatment with antibiotics or hospitalization within the last 4 weeks prior to enrollment or known active bacterial, viral (including SARS-CoV-2), fungal, mycobacterial, parasitic, or other infection (excluding fungal infections of nail beds) at study enrollment Evidence of any significant concomitant disease that could affect compliance with the protocol or interpretation of results Recent major surgery within 4 weeks prior to first study treatment administration, with the exception of protocol-mandated procedures (e.g., tumor biopsies and bone marrow biopsies)