KUALA LUMPUR, Malaysia—When President Barack Obama arrives in Malaysia on Friday, he will face a diplomatic dilemma: How, exactly, should a U.S. leader handle an important regional partner whose leader is engulfed in scandal?

Mr. Obama plans to spend three days in Kuala Lumpur for a pair of regional summits. It is the president’s interactions on the margins with Malaysian Prime Minister Najib Razak, however, that are likely to generate more political intrigue.

Mr. Najib sits at the center of a corruption probe relating to allegations that billions of dollars have disappeared from a Malaysian investment fund he started, and that $700 million tied to the fund ended up in the prime minister’s private accounts. Critics have been detained. A deputy prime minister who questioned the handling of the case has been fired. And the Malaysian government temporarily suspended two publications that were reporting on the controversy.

With investigations now under way in five countries and calls for the prime minister’s ouster continuing, Mr. Najib’s critics have urged Mr. Obama to keep his distance or at least publicly raise objections.

Malaysia, though, is a key strategic partner in the region, a counterbalance to China’s rise and one of the founding members of a sweeping Pacific trade accord. The Malaysian prime minister has denied wrongdoing or taking money for personal gain.