Axitinib and Bosutinib in Treating Patients With Chronic, Accelerated, or Blastic Phase Chronic Myeloid Leukemia

Inclusion Criteria:

* Diagnosis of Philadelphia chromosome positive (Ph+) (by cytogenetics or FISH) or BCR-ABL+ (by polymerase chain reaction \[PCR\]) CML in CP (cohort 1), AP (cohort 2) or BP (cohort 2)
* Patients should have failed (demonstrated resistance, intolerance or treatment discontinuation for any other reason of) at least 3 Food and Drug Administration (FDA)-approved TKIs if in CP (cohort 1), or at least 1 FDA-approved TKI if in AP (cohort 2); resistance will be defined as meeting the criteria for failure or warning by the European Leukemia Net (ELN); no prior therapy is necessary for patients in BP (cohort 2); patients in CP who have failed \< 3 TKIs, but are ineligible to receive other FDA-approved TKIs, may also be enrolled in cohort 1; at least 10 CP patients with the T315I mutation affecting the kinase domain of Bcr-Abl will be enrolled in cohort 1, as well as in the phase II portion of cohort 2
* Eastern Cooperative Oncology Group (ECOG) performance status of 0-2
* Total bilirubin =\< 1.5 x upper limit of normal (ULN) (unless due to Gilbert syndrome, in which case it should be =\< 3.0 x ULN)
* Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 x ULN
* Serum creatinine =\< 1.5 x ULN
* Patients must sign the Institutional Review Board (IRB)-approved informed consent document for this trial
* Reliable telephone access so as to be able to receive calls from an interactive voice response (IVR) system (only applicable to patients participating in the optional symptom burden assessment portion)
* Women of childbearing potential (WOCBP) must practice 2 effective methods of birth control during the course of the study; male patients who are partners of WOCBP should also practice an effective method of contraception: postmenopausal women must be amenorrheic for \>= 12 months to be considered of non-childbearing potential; women and men must continue birth control for the duration of the trial and \>= 3 months after the last dose of study drug; all WOCBP MUST have a negative pregnancy test prior to first receiving study medication(s)
* Patients should have discontinued therapy with imatinib, dasatinib, nilotinib, ponatinib, omacetaxine or other anti-leukemia therapy (except hydroxyurea) \>= 48 hours prior to start of study therapy and recovered from any toxicity due to these therapies to grade =\< 1; hydroxyurea may be received up to the time of enrollment and for the first 6 weeks of study treatment if necessary

Exclusion Criteria:

* Prior therapy with axitinib; prior therapy with bosutinib is allowed, except in the following circumstances: the subject is currently on bosutinib; bosutinib is the subject's most recent TKI for CML; the subject has a history of intolerance to bosutinib
* Active gastrointestinal conditions that are expected to impair absorption of orally administered medications
* Patients who currently have or have a history of the following within 6 months preceding study entry are not eligible: unstable angina (UA), myocardial infarction (MI), transient ischemic attack (TIA), stroke, deep vein thrombosis (DVT), acute peripheral or pulmonary arterial thromboembolism (PE); clinically significant ventricular arrhythmias (e.g., ventricular tachycardia, ventricular fibrillation, or torsades de pointes); New York Heart Association class III or IV heart failure
* Patients with active, uncontrolled psychiatric disorders including: psychosis, major depressive, and bipolar disorders
* Patients with uncontrolled hypertension (defined as sustained systolic blood pressure \> 160 mmHg or diastolic blood pressure \> 100 mmHg)
* Pregnant or breast-feeding women are excluded
* Inability to understand a written informed consent document
* Patients receiving anticoagulants that are unable to be discontinued
* Patients with active, uncontrolled infection
* Patients with a history of hypersensitivity to bosutinib or axitinib
* Patients on proton pump inhibitors, potent CYP3A or P-glycoprotein substrates, inhibitors or inducers a minimum 7 day period washout required unless discontinuation or substitution is not in the best interests of the patient as determined by the investigator; in instances where use of these agents is felt to be required for optimal management, inclusion of such patients should be discussed with the principal investigator (PI) and the rationale documented; these patients, if enrolled on study, may require dose modifications for both axitinib and bosutinib