A Study of TAK-981 Given With Rituximab in Adults With Relapsed or Refractory CD20-Positive Non-Hodgkin Lymphoma

This study is about a medicine called TAK-981 given with rituximab, used to treat adults with relapsed or refractory CD20-positive non-Hodgkin lymphoma. This study has 2 parts. The main aims of the study are: * To check for side effects from treatment with TAK-981 given with rituximab. * To check how much TAK-981 participants can tolerate. * To check if participants with diffuse large B-cell lymphoma or follicular lymphoma respond well to treatment. Participants will receive TAK-981 and rituximab in 21-day cycles. They will continue treatment for about 12 months unless their condition gets worse (disease progression), they cannot tolerate the treatment, or they leave the study for certain reasons.

The drug being tested in this study is called TAK-981 in combination with rituximab. The study will include a dose escalation phase (Phase 1) and an expansion phase in select non-Hodgkin lymphoma (NHL) indications (Phase 2). The study will enroll approximately 180 participants, approximately 35 participants in Phase 1 and approximately 145 participants in Phase 2. The participants with indolent or aggressive relapsed or refractory (r/r) NHL in Phase 1 will identify the maximum tolerated dose (MTD) and/or pharmacologically active dose (PAD). PAD can be defined retrospectively once MTD is reached and it can below MTD or coincide with it. In the dose escalation phase, the starting dose of TAK-981 will be 10 mg. The RP2D will be determined based on the available safety, preliminary pharmacokinetic (PK), pharmacodynamic information data, and after any early antitumor activity observed along with the statistical inference from the Bayesian logistic regression modeling (BLRM). Participants in the Phase 2 will be enrolled once the Phase 1 of the study is completed, and MTD and/or PAD is determined. Phase 2 will explore the efficacy and safety of TAK-981 in combination with rituximab in participants with select NHL types and indications. Participants in Phase 2 will be enrolled in one of the three treatment arms based on Cohorts: * Phase 2, Cohort A r/r DLBCL Progressed after CAR T-cell therapy:TAK-981+Rituximab * Phase 2, Cohort B:r/r DLBCL;no CAR T-cell Therapy;2-3 Prior Lines: Two Dose Levels of TAK-981+Rituximab * Phase 2, Cohort C:r/r FL;no CAR T-cell Therapy;2-3 Prior Lines: Two Dose Levels of TAK-981+Rituximab This multi-center trial will be conducted worldwide. The overall time to participate in this study is approximately 72 months. Participants will make multiple visits to the clinic, and will attend the end of treatment (EOT) visit 30 days after receiving their last dose of drug or before the start of subsequent systemic anticancer therapy, whichever occurs first for a follow-up assessment.