Exhibit 5.1

New York Menlo Park Washington DC São Paulo London Paris Madrid Tokyo Beijing Hong Kong

Davis Polk & Wardwell LLP Paseo de la Castellana, 41 28046 Madrid 91 768 9600 tel 91 768 9700 fax

May 13, 2015

Telefónica, S.A.

Distrito Telefónica, Ronda de la Comunicación, s/n

28050 Madrid

Spain

Telefónica Emisiones, S.A.U.

Distrito Telefónica, Ronda de la Comunicación, s/n

28050 Madrid

Spain

Ladies and Gentlemen:

We are acting as special United States counsel to Telefónica, S.A., a sociedad anónima organized under the laws of the Kingdom of Spain (“Telefónica”), and Telefónica Emisiones, S.A.U., a sociedad anónima unipersonal organized under the laws of the Kingdom of Spain (“Telefónica Emisiones”), in connection with the Registration Statement on Form F-3 (the “Registration Statement”) filed today by Telefónica and Telefónica Emisiones with the United States Securities and Exchange Commission pursuant to the United States Securities Act of 1933, as amended (the “Securities Act”), for the registration and sale from time to time of an indeterminate amount of the following securities: (i) ordinary shares of Telefónica, including in the form of American Depositary Shares; (ii) rights to subscribe for ordinary shares of Telefónica; and (iii) debt securities of Telefónica Emisiones (the “Debt Securities”) and the guarantee of each such series of Debt Securities by Telefónica (the “Guarantee”).

The Debt Securities may be issued from time to time pursuant to an Indenture (the “Indenture”) dated May 22, 2012 among Telefónica Emisiones, Telefónica and The Bank of New York Mellon, as trustee (the “Trustee”). The Guarantee may be issued under a separate guarantee, the form of which is included in the Indenture, to be entered into by Telefónica prior to the issuance of the respective Debt Securities.

We, as your counsel, have examined originals or copies of such documents, corporate records, certificates of public officials and other instruments as we have deemed necessary or advisable for the purpose of rendering this opinion.

In rendering the opinions expressed herein, we have, without independent inquiry or investigation, assumed that (i) all documents submitted to us as originals are authentic and complete, (ii) all documents submitted to us as copies conform to authentic, complete originals, (iii) all documents filed as exhibits to the Registration Statement that have not been executed will

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conform to the forms thereof, (iv) all signatures on all documents that we reviewed are genuine, (v) all natural persons executing documents had and have the legal capacity to do so, (vi) all statements in certificates of public officials and officers of Telefónica Emisiones and Telefónica that we reviewed were and are accurate and (vii) all representations made by Telefónica Emisiones and Telefónica as to matters of fact in the documents that we reviewed were and are accurate.

Based upon the foregoing, and subject to the additional assumptions and qualifications set forth below, we advise you that, in our opinion, assuming that (i) the Indenture and any supplemental indenture to be entered into in connection with the issuance of any Debt Securities have been duly authorized, executed and delivered by Telefónica Emisiones, Telefónica and the Trustee; (ii) the specific terms of a particular series of Debt Securities and the related Guarantee have been duly authorized and established in accordance with the Indenture; and (iii) such Debt Securities and the related Guarantee have been duly authorized, executed, authenticated, issued and delivered in accordance with the Indenture and the applicable underwriting or other agreement against payment therefor, such Debt Securities will constitute valid and binding obligations of Telefónica Emisiones and the related Guarantee will constitute valid and binding obligations of Telefónica, enforceable in accordance with their terms, subject to applicable bankruptcy, insolvency and similar laws affecting creditors’ rights generally, concepts of reasonableness and equitable principles of general applicability, provided that we express no opinion as to the enforceability of any waiver of rights under any usury or stay law or the validity, legally binding effect or enforceability of any provision that permits holders to collect any portion of stated principal amount upon acceleration of such Debt Securities to the extent determined to constitute unearned interest.

In connection with the opinion expressed above, we have assumed that at or prior to the time of the delivery of any Debt Securities or Guarantee, as the case may be, (i) the Board of Directors of Telefónica Emisiones and the Board of Directors of Telefónica, as the case may be, shall have duly established the terms of such Debt Securities and Guarantee, as the case may be, and duly authorized the issuance and sale of such Debt Securities and Guarantee, as the case may be, and such authorization shall not have been modified or rescinded; (ii) each of Telefónica Emisiones and Telefónica is, and shall remain, validly existing as a corporation under the laws of Spain; (iii) the Registration Statement shall have become effective and such effectiveness shall not have been terminated or rescinded; (iv) the Indenture, the Debt Securities and the Guarantee are each valid, binding and enforceable agreements of each party thereto (other than as expressly covered above in respect of Telefónica Emisiones and Telefónica); and (v) there shall not have occurred any change in law affecting the validity or enforceability of such Debt Securities and Guarantee, as the case may be. We have also assumed that the execution, delivery and performance by Telefónica Emisiones of any Debt Security and by Telefónica of any Guarantee, whose respective terms are established subsequent to the date hereof (a) require no action by or in respect of, or filing with, any governmental body, agency or official and (b) do not contravene, or constitute a default under, any public policy, any provision of applicable law or regulation or any judgment, injunction, order or decree or any agreement or other instrument binding upon Telefónica Emisiones or Telefónica, as the case may be.

We are members of the Bar of the State of New York and the foregoing opinion is limited to the laws of the State of New York and the federal laws of the United States. Insofar as the foregoing opinion involves matters governed by the laws of Spain, we have relied, without independent inquiry or investigation, on the opinion of Pérez-Llorca Abogados, S.L.P. y Cía., S.Com.P. dated May 13, 2015, filed as Exhibit 5.2 to the Registration Statement.

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We hereby consent to the filing of this opinion as an exhibit to the Registration Statement referred to above and further consent to the reference to our name under the caption “Legal Matters” in the prospectus dated May 13, 2015 which is a part of the Registration Statement. In giving this consent, we do not admit that we are in the category of persons whose consent is required under Section 7 of the Securities Act.

Very truly yours,

/s/ Davis Polk & Wardwell LLP