This is an open-label, single-arm, phase I trial. It is designed with a conservative dose escalation plan to ensure patient's safety and with a strong translational component to inform if target inhibition is achieved. With concerns regarding safety, based on extensive available pharmacokinetic data and clinical efficacy experience, bosutinib will be given 5-days in a row followed by 2 days rest in a weekly basis, instead of daily.

The protocol will enroll patients per 3+3 escalation design. The Dose Limiting Toxicity (DLT) observation period is 28 days. At the end of DLT observation period of each cohort of 3 patients, decision will be made regarding further escalation or de-escalation according to this plan. Once the MTD of the combination is reached, the safety data will be analyzed. There will be no dose reductions during DLT observation period. Dose reduction within patients (individually) is allowed after the 4-week DLT observation period. Treatment in this phase I trial will be administered until there is disease progression or unacceptable toxicity.

Inclusion Criteria:

1. Signed informed consent obtained prior to any study specific assessments and procedures.

2. Age ≥18 years

3. Premenopausal and postmenopausal women

4. Biopsy proven diagnosis of ER and/or PR positive, HER2 negative, advanced breast cancer (locoregionally recurrent or metastatic disease), either from the primary or a metastatic site.

   ER, PR and HER2 measurements should be performed according to institutional guidelines, in a CLIA-approved setting.

   Breast cancer is ER-positive and/or PR-positive tumor (≥1% positive stained cells) based on local CLIA-certified laboratory results

   HER2-negative breast cancer:

   Cut-off values for positive/negative staining should be in accordance

5. A formalin-fixed paraffin-embedded (FFPE) tumor tissue block from diagnostic biopsy must be transmitted to MedStar Georgetown University Hospital Pathology Department repository and confirmation of receipt must be available prior to initiation of treatment on study.

6. ECOG performance status 0-1

7. Must have received no more than 3 lines of chemotherapy for the treatment of breast cancer and be progressive on at least one aromatase inhibitor and one CDK 4/6 inhibitor.

8. Pregnancy must be ruled out Serum or urine pregnancy test must be negative within 14 days of treatment start in women of childbearing potential.

   Pregnancy testing does not need to be pursued in patients who are judged as postmenopausal before enrollment, or who have undergone bilateral oophorectomy, total hysterectomy, or bilateral tubal ligation.

   Patients may be considered postmenopausal in case that one of the following criteria applies (Section 5.4.1.2):

   Prior bilateral oophorectomy, OR Age ≥ 60 years, OR Age < 60 years with intact uterus and amenorrhoeic for ≥ 12 consecutive months prior to chemotherapy and/or endocrine therapy exposure, OR Age < 60 years hysterectomized and FSH and plasma estradiol levels in the post-menopausal range according to local policies prior to chemotherapy and/or endocrine therapy exposure

9. Willingness to undergo adequate contraception if childbearing potential Women of childbearing potential must use adequate contraception for the duration of protocol treatment and for 3 months after the last treatment with palbociclib/bosutinib.

   Adequate contraception is defined as one highly effective form (i.e. abstinence, (fe)male sterilization) OR two effective forms (e.g. non-hormonal IUD and condom / occlusive cap with spermicidal foam / gel / film / cream / suppository).

10. Patients must be able and willing to swallow and retain oral medication

11. Absolute neutrophil count (ANC) ≥ 1,500/mm^3

12. Platelets ≥ 100,000/mm^3

13. Hemoglobin ≥ 9 g/dL

14. AST and/or ALT ≤3 x ULN

15. Alkaline phosphatase ≤2.5 x ULN (≤5.0 x ULN if bone metastases present)

16. Total serum bilirubin ≤1.5 x ULN

17. Serum creatinine within normal institutional limits or creatinine clearance ≥ 50 mL/min/1.73 m^2 for patients with serum creatinine levels above institutional ULN.

18. Resolution of all acute toxic effects of prior therapy, including radiotherapy to grade ≤1 (except toxicities not considered a safety risk for the patient) and recovery from surgical procedures.

19. Patients who are willing and able to comply with scheduled visits, treatment plan, laboratory tests, and other study procedures.

Exclusion Criteria:

1. Concurrent therapy with other Investigational Products.
2. Current use of food or drugs known to be potent inhibitors or inducers of CYP3A4
3. Known hypersensitivity to fulvestrant, palbociclib or bosutinib, or to any of their excipients.
4. Uncontrolled intercurrent illness including (active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, diabetes, pulmonary embolism in the past 6 months, or psychiatric illness/social situations that would limit compliance with study requirements).
5. Active uncontrolled or symptomatic brain metastases. Previously treated and clinically stable, as per Investigator's judgment, brain metastases are permitted.
6. Unable to comply with study requirements
7. Presence of a condition that would interfere with enteric absorption of palbociclib/bosutinib.
8. Pregnant women, or women of childbearing potential without a negative pregnancy test (serum or urine) within 14 days prior to starting treatment on study Breastfeeding must be discontinued prior to study entry.
9. Patients on combination antiretroviral therapy, i.e. those who are HIV-positive (potential for pharmacokinetic interactions or increased immunosuppression with palbociclib).
10. Patients with clinically significant history of liver disease, including viral or other known hepatitis, current alcohol abuse, or cirrhosis, etc.
11. Patients on chronic anticoagulation (fulvestrant is IM injection)