The Championship outfit have been in the market for a new manager since Michael Appleton switched to Blackburn Rovers last month.
However, their search will now continue after Davies rejected their advances on Tuesday.
"I would like to place on record that the club met and discussed the manager's position with Billy Davies on Monday evening," Oyston said in a statement on the club's official website. "Contrary to speculation, I can confirm that the meeting concluded with both parties agreeing to speak again on Tuesday.
"I spoke with Billy on two occasions yesterday and he has decided to follow another opportunity, which is a decision we fully appreciate and understand."
Oyston went on to confirm that the vacancy had only been offered to just one person, but Sean O'Driscoll decided to take charge of league rivals Bristol City.
"It has been well documented that we have approached a number of people regarding our managerial vacancy, in order to conduct preliminary discussions and ascertain interest from either party," he added.
"We will continue with this process and we will continue to do so in the correct manner. However, we have only offered the position to one person and, unfortunately, that person decided to accept a position at Bristol City."
It is believed that Davies turned down Blackpool because he is being lined up for a second stint as manager of Nottingham Forest.