Spurs were 10 points behind fourth-placed Arsenal in the Premier League at the end of last season, but King believes that Pochettino's side are not out of the running for a spot in Europe's elite competition ahead of the new campaign.
"There are seven teams who can finish in the top four," the former England defender told the Daily Star. "There are two or three, maybe four, who believe they can win the league. We are not one of them at the moment.
"We need a bit of time. We have a new manager and a new style of play. It's going to be tough to break into the top four, but if we have a good season we are capable of doing that.
"It's not easy but the players are capable of doing that. They just have to believe it. We have a very good manager, so you never know."
Spurs begin the new season against West Ham United at Upton Park on Saturday.