BFR Bbva Banco Frances S.A.

SECURITIES AND EXCHANGE COMMISSION

Washington, D.C. 20549

FORM 6-K

Report of Foreign Issuer

Pursuant to Rule 13a-16 or 15d-16

of the Securities Exchange Act of 1934

For the month of June 2016

Commission File Number: 001-12568

BBVA French Bank S.A.

(Translation of registrant’s name into English)

Reconquista 199, 1006

Buenos Aires, Argentina

(Address of principal executive offices)

Indicate by check mark whether the registrant files or will file annual reports under cover of Form 20-F or Form 40-F:

Form 20-F   x                 Form 40-F   ¨

Indicate by check mark if the registrant is submitting the Form 6-K in paper as permitted by Regulation S-T Rule 101(b) (1):

Yes   ¨                  No   x

Indicate by check mark if the registrant is submitting the Form 6-K in paper as permitted by Regulation S-T Rule 101(b) (7):

Yes   ¨                  No   x

Indicate by check mark whether by furnishing the information contained in this Form, the Registrant is also thereby furnishing the information to the Commission pursuant to Rule 12g3-2(b) under the Securities Exchange Act of 1934:

Yes   ¨                  No   x

If “Yes” is marked, indicate below the file number assigned to the registrant in connection with Rule 12g3-2(b): N/A

BBVA Banco Francés S.A.

TABLE OF CONTENTS

June 8, 2016

Securities and Exchange Commission

RE: Financial Consumers Association for Civil Defense against BBVA Banco Francés S.A

We address to you in our capacity as Market Relations Authority of BBVA Banco Frances S.A. in order to report that the Bank has been served of a class action lawsuit filed by the “Financial Consumers Association for Civil Defense” Association.

The Association on behalf of consumers, claims that certain products such as ATM theft insurance, personal accident and home insurance are granted by the Bank without the express consent of customers.

The Bank is currently analyzing the content of the demand and defining the course of action, believing that this action could not have a significant impact on the Bank’s assets.

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