His father has not been introduced. The anime suggests that his father was once a Pokémon Trainer who began his journey in Pallet Town, and that he is still living.
So, it's not known if he does have a father that is alive/he knows of (dangit trachy xP), only a theory.
However, in the games, the protagonist does have a father.
In R/S/E, his father is the Normal-type Gym Leader. In the Other games, when you talk to his mother, sometimes, she will tell the protagonist that she reminds him of his father, who was a Pokemon Trainer that left on his journey a long time ago.
Source: Bulbapedia + Experience