WASHINGTON—Michael Cohen, who has served as President Donald Trump’s personal attorney, made an overture to provide consulting services to Ford Motor Co. in January 2017 but was quickly rebuffed, people familiar with the matter said.

Mr. Cohen, touting his proximity to the president, contacted Ford’s office in Washington, D.C.—an approach that Special Counsel Robert Mueller learned about in the course of his investigation, the people said.

Mr. Mueller’s team has since requested information from Ford about the outreach, including emails and records, and has interviewed Ford’s head of government affairs, Ziad Ojakli, the people said.

The pitch was made in an informal phone call and the exchange was short. Mr. Ojakli rejected the offer and had no further conversations about it, the people said.

It is unclear when Mr. Mueller sought information about the overture from Mr. Cohen. Mr. Mueller is looking into possible collusion between the Trump campaign and Russian attempts to influence the 2016 elections. He is also examining whether the president sought to obstruct justice. Mr. Trump has denied collusion and obstruction of justice, and Moscow has denied meddling in the election.

Representatives for Mr. Cohen didn’t respond to a request for comment; Mr. Mueller’s spokesman declined to comment.

Mr. Cohen’s consulting work has drawn scrutiny from federal prosecutors.

He set up a company, Essential Consultants LLC, that he used to make a payment to the former adult-movie actress Stephanie Clifford—known as Stormy Daniels—in exchange for her silence about an alleged affair with President Donald Trump.

Last year, Mr. Mueller’s office sought information from the pharmaceutical company Novartis AG and the telecommunications firm AT& T Inc. dealing with payments they made to Essential Consultants in exchange for Mr. Cohen’s advice.

Mr. Cohen’s company has become a focal point of an investigation led by the U.S. Attorney’s office in Manhattan into whether Mr. Cohen committed bank fraud and other crimes, people familiar with the matter said.

In April, FBI agents searched Mr. Cohen’s office, home and hotel room as part of an investigation by the Manhattan U.S. Attorney’s office, which had received a referral from Mr. Mueller.

—Rebecca Ballhaus contributed to this article.

Write to Peter Nicholas at peter.nicholas@wsj.com and Christina Rogers at christina.rogers@wsj.com