UNITED STATES

SECURITIES AND EXCHANGE COMMISSION

WASHINGTON D.C. 20549

FORM 6-K

REPORT OF FOREIGN PRIVATE ISSUER

PURSUANT TO RULE 13a-16 OR 15d-16

OF THE SECURITIES EXCHANGE ACT OF 1934

July 2026

Commission File Number: 001-43208

The Metals Royalty Company Inc.

1900 Dome Tower

333 7^th Ave SW

Calgary, AB, T2P 2Z1

British Columbia, Canada

(403) 984-1941

(Address of principal executive offices)

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

Form 20-F x         Form 40-F ¨

INFORMATION CONTAINED IN THIS REPORT ON FORM 6-K

On July 13, 2026, The Metals Royalty Company Inc (the “Company”) issued a press release titled “The Metals Royalty Company Announces Completion of First Production Blast at Mesabi Metallics.” A copy of the press release is furnished here as Exhibit 99.1 and shall not be deemed “filed” for purposes of Section 18 of the Securities Exchange Act of 1934, as amended (the “Exchange Act”), or otherwise subject to the liabilities of that section, nor shall they be deemed incorporated by reference in any filing under the Securities Act of 1933, as amended, or the Exchange Act.

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SIGNATURE

Pursuant to the requirements of the Securities Exchange Act of 1934, the registrant has duly caused this report to be signed on its behalf by the undersigned, thereunto duly authorized.

Date: July 14, 2026