Moon, a former leader of the largest party in parliament, garnered 34.9 percent in the survey conducted from Monday to Friday last week, up 0.5 percentage point from the previous week, Realmeter said.
Ahn Cheol-soo of the centrist People's Party jumped 6.1 percentage points to 18.7 percent to second place, beating other major candidates.
At the same time, the voice for a single candidacy agreement grows among conservative supporters.