Meanwhile, Arsene Wenger's side missed out on Champions League football with a fifth-placed finish but ended the season by lifting their third FA Cup in four years.
"There is always one odd year where things can change, but the most important thing for us is competing against the other teams for the Premier League title and for the glory," Cech told Sky Sports.
"We are not competing against Tottenham Hotspur. Two years ago when we finished above them we were disappointed because we didn't win the title.
"Our target is to win the title and be successful during the season, not necessarily looking at what Tottenham are doing."
Pochettino's Spurs have challenged for the league title over the last two seasons, while their recent victories over Real Madrid and Borussia Dortmund have placed them among the top clubs in the Champions League.
But having failed to win a major trophy since the League Cup in 2008, Cech believes winning silverware is necessary for Spurs to mark their recent progress.
"There was always something missing but I would say the difference was when Pochettino came," the 35-year-old from Czech Republic added.
"They've been there in the last few years so now I think they have to make sure they win something to show the progression."
Arsenal, currently sixth in the standings, host third-placed Tottenham at the Emirates Stadium in the league on Saturday.
(Reporting by Hardik Vyas in Bengaluru; Editing by Christian Radnedge)
