The Justice Department’s internal watchdog, close to concluding its inquiry into steps the FBI took in its probe of Trump campaign associates and Russia beginning in 2016, is homing in on whether the agency continued surveillance of one associate despite questions about a key source’s credibility, according to people familiar with the matter.

In the closely watched probe of the counterintelligence inquiry that later morphed into special counsel Robert Mueller’s examination of Russia’s interference in the 2016 election, the...