Opposition supporters celebrated in the streets of the capital Dakar on Sunday after local media channels started announcing polling station tallies that put Faye, comfortably ahead of his main rival, Amadou Ba.
Several other opposition contenders in Sunday's election conceded defeat to Faye during the night.
Official results are expected to be announced by the Dakar appeals court on Friday.
Many hope the delayed vote will bring stability and an economic boost to the West African nation after three years of unprecedented political turbulence.
Ba is the candidate backed by outgoing President Macky Sall, who is stepping down amid a drop in popularity after two terms in office marred by economic hardship and violent anti-government protests.
Ba said in a statement late on Sunday that celebrations of Faye's victory were premature and that official results would show a second round of voting was needed.