Boeing Co.’s board will add a director with safety expertise and adopt other internal measures under a proposed $237.5 million agreement to settle a shareholder lawsuit that alleged lax board oversight led to the 737 MAX disasters.

According to a proposed settlement filed Friday in Delaware’s Court of Chancery, the plane maker has agreed to add another director to its 12-member board with aviation, engineering or product-safety oversight experience. Boeing also agreed to ensure at least three directors have such expertise....

Boeing Co.’s board will add a director with safety expertise and adopt other internal measures under a proposed $237.5 million agreement to settle a shareholder lawsuit that alleged lax board oversight led to the 737 MAX disasters.

According to a proposed settlement filed Friday in Delaware’s Court of Chancery, the plane maker has agreed to add another director to its 12-member board with aviation, engineering or product-safety oversight experience. Boeing also agreed to ensure at least three directors have such expertise. The company earlier this year added to the board a former airline pilot and a former executive from an engine-maker.

A new ombudsman program Boeing has agreed to adopt would provide an additional channel for company employees who act on behalf of federal air-safety regulators to raise concerns, including over transparency and undue pressure, according to the settlement agreement.

Such issues have been a focus of the Federal Aviation Administration, Boeing’s primary aviation regulator, and congressional investigators. The proposed settlement, which is subject to approval by the judge overseeing the case, also requires Boeing to amend its bylaws to memorialize the separation of the CEO and board chairmanship, a step the company took in 2019.

Plaintiffs in the case said the governance changes would enhance Boeing’s safety focus, protect shareholder value and send a message to corporate directors.

“It is our hope, moving forward, that the reforms agreed to in this settlement will help safeguard Boeing and the flying public against future tragedy and begin to restore the company’s reputation,” said New York State Comptroller Thomas DiNapoli, co-plaintiff in the case along with the Fire and Police Pension Association of Colorado.

Boeing said that since two 737 MAX jets crashed, the company has taken significant steps to strengthen its commitment to aviation safety. “Today’s proposed settlement builds on those actions with additional oversight and governance reforms that will further advance safety and quality in the work that we do,” Boeing said Friday.

Two of Boeing’s 737 MAX jets crashed in late 2018 and early 2019, taking 346 lives. The company later shook up its board and management and changed how it handles safety, including adding a board committee focused on the topic.

The proposed settlement includes no admission of fault or wrongdoing on the part of the current and former directors, and notes the defendants deny the lawsuit’s allegations. The monetary portion of the settlement will be paid to the corporation by the directors’ insurance providers.

The plaintiffs attorneys in the case are expected to seek $29.7 million in fees and expenses, or 12.5% of the monetary portion of the settlement, according to court documents filed in the case.

If approved by the judge, the agreement would mark the end of one of the various legal proceedings Boeing has faced since the MAX crashes. Earlier this year it agreed to pay $2.5 billion to settle the U.S. Justice Department’s criminal investigation into the 737 MAX. It continues to face lawsuits from crash victims’ families.

Among its claims, the lawsuit alleged Boeing’s board failed to challenge then-CEO Dennis Muilenburg

over 737 MAX safety issues or his campaign to counter negative news reports between the two fatal accidents.

The suit also claimed Boeing’s Mr. Calhoun, who previously served as lead director and chairman, exaggerated to journalists the extent of directors’ safety oversight.

Boeing had argued the plaintiffs’ suit portrayed an incomplete and misleading picture of directors’ safety oversight. The company has also said the plaintiffs misrepresented Mr. Calhoun’s public comments about board oversight.

The Boeing case is different from a securities-fraud lawsuit through which plaintiffs themselves might recoup investment losses. Known as a shareholder derivative complaint, such legal actions seek to hold company officials responsible for alleged missteps and can result in corporations receiving monetary awards and adopting governance changes.

Write to Andrew Tangel at Andrew.Tangel@wsj.com