Deferasirox in Treating Patients With Very Low, Low, or Intermediate-Risk Red Blood Cell Transfusion Dependent Anemia or Myelodysplastic Syndrome

Inclusion Criteria: * Capable of giving written informed consent prior to any study-specific procedures * Diagnosis of MDS as defined by the World Health Organization (WHO) diagnostic criteria * Have very low, low or intermediate-risk disease by the Revised International Prognostic Scoring System (IPSS-R) * Baseline serum ferritin level \>= 100 ng/mL * Have an Eastern Cooperative Oncology Group (ECOG) performance status of 0-2 * Anemia defined as: hemoglobin =\< 10.0 g/dL * Bilirubin =\< 1.5 times upper limit of normal (ULN) * Alanine aminotransferase (ALT), and aspartate aminotransferase (AST) =\< 3.5 times ULN * Serum creatinine =\< 1.5 x ULN * Estimated glomerular filtration rate (GFR) \> 40 mL/min * Males and females with reproductive potential must agree to use medically approved contraceptive precautions during the study and for 3 months following the last dose of deferasirox * Women of child-bearing potential, defined as all women physiologically capable of becoming pregnant, unless they are using effective methods of contraception during dosing of study treatment; effective contraception methods include: * Placement of an intrauterine device (IUD) or intrauterine system (IUS) * Barrier methods of contraception: condom or occlusive cap (diaphragm or cervical/vault caps) with spermicidal foam/gel/film/cream/vaginal suppository * Total abstinence or (when this is in line with the preferred and usual lifestyle of the subject); periodic abstinence (e.g., calendar, ovulation, symptothermal, post-ovulation methods) and withdrawal are not acceptable methods of contraception * Female sterilization (have had surgical bilateral oophorectomy with or without hysterectomy) or tubal ligation at least six weeks before taking study treatment; in case of oophorectomy alone, only when the reproductive status of the woman has been confirmed by follow up hormone level assessment * Male sterilization (at least 6 months prior to screening); for female subjects on the study, the vasectomized male partner should be the sole partner for that subject * Women are considered post-menopausal and not of child bearing potential if they have had 12 months of natural (spontaneous) amenorrhea with an appropriate clinical profile (e.g. age appropriate, history of vasomotor symptoms) or have had surgical bilateral oophorectomy (with or without hysterectomy) or tubal ligation at least six weeks ago; in the case of oophorectomy alone, only when the reproductive status of the woman has been confirmed by follow up hormone level assessment is she considered not of child bearing potential; sexually active males must use a condom during intercourse while taking drug and for 28 days after stopping study medication and should not father a child in this period; a condom is required to be used also by vasectomized men in order to prevent delivery of the drug via seminal fluid * Females with childbearing potential\* must have had a negative urine or serum pregnancy test =\< 7 days before the first dose of deferasirox and must also not be breastfeeding * Reliable and willing to make themselves available for the duration of the study and are willing to follow study procedures Exclusion Criteria: * If the patient is currently receiving erythroid stimulating agents (ESA) with plans to continue during study, less than 2 months duration of ESA prior to starting study drug and no dose escalation within 2 months of start of study drug * If the patient is being treated with granulocyte-colony stimulating factor (GCSF) and/or a TPO-mimetic (for example, eltrombopag or romiplostim) with plans to continue during the study: Less than 2 months duration of GCSF or the TPO-mimetic treatment prior to starting study drug; or GCSF and/or TPO-mimetic has been added to ESA therapy within 2 months of start of study drug * If patient is being treated with lenalidomide with plans to continue during the study: Stable dose for less than 3 months prior to start of study drug * If patient is being treated with hypomethylating agents (HMA) (for example, azacitidine or decitabine) with plans to continue during the study: Stable dose for less than 6 months prior to start of study drug * Currently enrolled in, or discontinued within the last 14 days from a clinical trial involving an investigational product or non-approved use of a drug, or concurrently enrolled in any other type of medical research judged not to be scientifically or medically compatible with this study * Presence of \>= 10% blast by morphologic examination of bone marrow aspirate or biopsy * Platelets =\< 50,000 * Microcytosis on screening blood cell count (CBC) (mean corpuscular volume \[MCV\] \< 81 fL) * Active gastrointestinal (GI) ulceration or hemorrhage * Have a serious preexisting medical condition that, in the opinion of the investigator would preclude participation in the study (for example a GI disorder causing clinically significant symptoms such as nausea, vomiting, and diarrhea, or malabsorption syndrome) or that would result in a life expectancy of less than 1 year * Known hypersensitivity to deferasirox * History of non-transfusional hemosiderosis * Prior hematopoietic stem cell transplant for the diagnosis of MDS * A second primary malignancy that in the judgment of the principal investigator (PI) or designee may affect the interpretation of results * Have an active fungal, bacterial, and/or known viral infection including human immunodeficiency virus (HIV) or viral (A, B, or C) hepatitis * Currently using aluminum-containing antacid products * History of clinically significant auditory or ocular toxicity with ICT