BOSTON—The man behind the scheme to help wealthy students get into elite colleges by cheating on tests and faking athletic credentials advised some families to falsely claim students were racial minorities, according to people familiar with the situation.

A son of Marjorie Klapper, a parent scheduled to plead guilty Friday for participating in the scheme, was incorrectly listed on his Common Application as being black and Hispanic, the people said. William “Rick” Singer, the college counselor who has agreed to plead guilty...