With 31 percent of the estimated votes counted, Warnock led with 63 percent to Walker's 37, according to Reuters.
The final result is expected to be presented later in the morning, Swedish time.
Should Warnock win, it would give Democrats a 51-seat majority in the 100-seat Senate, which would make it somewhat easier to advance Biden's nominees to judicial and administrative posts.
Most legislation would still require Republican support.
A Republican victory, on the other hand, would keep the Senate at its current 50-50 split, with Democrats in control thanks to Vice President Kamala Harris' tie-breaking vote.