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