With nearly 40 million votes in Friday's hard-fought poll counted so far on Saturday, Rouhani was leading the race with 56.88 percent, the report said quoting the interior ministry.
Raisi, his closest rival, won 38.55 percent of the vote.
Mostafa Mirsalim, a conservative, got 1.13 percent, while reformist Mostafa Hashemitaba got 0.52 percent.
A big turnout on Friday, estimated at more than 40 million out of 56 million registered voters, led to the vote being extended by several hours.