"Mr. Nicolás Maduro, respect what all Venezuelans have decided. You and your government should step aside. I am ready to engage in a dialogue", he says in a video clip published on social media.
Since the election, both 74-year-old González Urrutia and his opposition colleague María Corina Machado have been in hiding after prosecutors launched criminal investigations against both of them.
On Saturday, Machado chose to appear in the capital Caracas in connection with the opposition's massive protests, which were held both in several locations around Venezuela and abroad.
Both Maduro and the opposition have claimed victory in the election on July 28, and the unrest that has followed has so far claimed 25 lives. Nearly 200 people have been injured and over 2,400 have been arrested.
Maduro's claim to electoral victory has been questioned by several Latin American countries, the US, and the EU.
According to the opposition, protocols collected directly from over 80 percent of the polling stations show that their candidate Edmundo González Urrutia won the election with a significant majority.
Nicolás Maduro portrays the opposition's claims of being the election winner as a coup attempt and has sworn to crush all resistance. Arrest warrants have been issued against opposition leaders González Urrutia and María Corina Machado.
The opposition, several Latin American countries, and the US are demanding that Venezuela present a detailed and verifiable election result. The regime has refused, citing that it is impossible. President Maduro has claimed that the electoral authority has been hacked.
The question of who actually won the presidential election has now been referred to the country's highest court, which is criticized by critics and the opposition as being loyal to Maduro.
Sources: Efe, El País, and others.