Mauricio Pochettino urged caution over Tottenham’s title prospects but admitted his team could rival Liverpool and Manchester City for the Premier League crown providing they maintain the form that demolished Everton 6-2 at Goodison Park.

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Spurs inflicted the heaviest defeat of Marco Silva’s Everton tenure with an outstanding display that featured two goals apiece for Harry Kane and Son Heung-min. The South Korea forward was applauded off by the home fans, such was his impact on a chastening afternoon for Everton and another demonstration of why Pochettino is coveted by Manchester United.

The Argentinian coach again refused to discuss “rumours” about a potential move to Old Trafford next summer and, though wary of talking up Spurs’s title prospects, claimed consistency could make them a reality.

Pochettino said: “I still believe Liverpool and Manchester City are the real contenders and favourites to win the Premier League. We are there but so are Chelsea, Arsenal and Manchester United. I think there is still a long way for me to say we are real contenders. Football is about being consistent, of course.

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“If we are able to be consistent and play in the way we did tonight I think yes, maybe we could then be a real contender but there is still a long way. We are involved in four competitions. I think today we surprised everyone with the energy and quality we showed today which was amazing after playing Arsenal on Wednesday.”

The Spurs manager said he was “proud” of the performance and that his team are getting closer to challenging for the championship. He added: “If you compare the result with the past of Tottenham then yes we were consistent, but not enough to be a real contender. Two seasons ago when we were fighting with Leicester … the gap was so big with different teams that won the Premier League: Chelsea or Manchester City last season. Now we are close but we need to be cautious, go step by step, game by game.

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“We need to be conscious about that, we need to fight a lot, and there is still a long way for work and to be consistent. Being involved in four competitions will be tough but we are going to try. The mentality today was clear and we need to always think about the next game and try to win every game and to be in the best condition to fight for big things.”

Spurs lost Dele Alli to a hamstring injury at half-time after the midfielder was on the receiving end of a bad challenge from Everton goalkeeper Jordan Pickford. “I didn’t see it,” said Pochettino. “I am very curious to see the action, I think it wasn’t a great action. I don’t want to say anything … but I think it was a tough action. We will see if he is injured. I think he is tired, tight in his hamstring, but we hope it is not a big issue.”