Spurs are 16/1 outsiders with Sky Bet to overturn a seven-point deficit on the top four between now and the end of the season.
Defeats to Chelsea and Arsenal in their past two Premier League games have done little for Tottenham's chances, and the first of those prompted manager Sherwood to publicly criticise his players' character.
Spurs have not won since but, speaking to Sky Sports News' Weekend Warm-up programme ahead of Sunday's game against Southampton, Kaboul backed Sherwood's assessment - and his potential to become a "very top manager".
He said: "(The manager) played with his heart and he wants to bring back this culture to us.
"Some of us have it already but we need 11 players with the same mentality on the pitch each game, which sometimes is hard.
"The ability is there, the quality is there but sometimes we are being too nice to each other. We can't have any more mistakes until the end of the season, we have to avoid that."
Kaboul was part of of the squad which did qualify for the Champions League in 2012, and he said: "I've been here long enough to understand and to know the club, what the club wants. I think we can always do better for Tottenham.
"What we want is Champions League football and we can do it. We have a lot of quality in the dressing room and we'll work hard to get there.
"We have to take points all the time and we are going to try. It's going to be tough but we've done this before."
Watch Tottenham v Southampton live this Sunday. Coverage begins at 12.30pm on Sky Sports 1