There have been many rumors. The top 3 are either Giovanni of Team Rocket, Silver, or Proffesor Oak. I personally belieave its either Giovanni or Oak. Look at the facts.
Giovanni
The proof: In Pokémon Live, Delia admitted her dark past. She used to go out with Giovanni back when he was starting a gang that would later become Team Rocket. Soon afterwards, she left him, married the mythical Mr. Ketchum and became the innocent little homemaker that she is today. But if Delia and Giovanni were a couple, couldn't that also mean that they might have had a son? A son that wasn't born until after Delia was married to Mr. Ketchum? Could Ash really be the heir to the Team Rocket empire?
The argument: Sorry, too many plot holes in that story. First of all, Giovanni didn't start Team Rocket. His mother, known as Madame Boss, was the leader before he was. Secondly, Pokémon Live is purely American, so it has no meaning in terms of the Japanese cartoon. The cartoon Delia seems to have no knowledge of Giovanni. Also, there isn't enough of a resemblance between Giovanni and Ash. The only one seems to be that they both have black hair. The facial structure and voice are too different for Giovanni to be the father.
Professor Oak
PROF. OAK
Samuel Oak has been living so close to the Ketchum residence all these years, and a Mr. Ketchum has never been spotted in the series. Could it be that Professor Oak is the father?
The proof: LOOK AT HIM. Now imagine him several years younger. His hair color would be either black or a very dark brown, as you can see in the eyebrows. They have almost the same hairstyle. Oak seems to be close to Ash's mom... a little too close... and they both have tree names. Professor Oak also seems to favor Ash over his own grandson, Gary! Could it be because (dun dun DUN!) Samuel Oak is the father of Ash Ketchum?! Why, yes it could! Also, as if we didn't have enough proof, Gary told Ash that Professor Oak used to be a great Pokémon trainer. Well, Delia also said the same thing about Ash's father. Hmmm...
Here is a Link to Ash being aged 40 years. See a resemblance?
Prof. Oak is the father!