The 31-year-old forward linked up with the Blues in January as a stint with AC Milan was cut short.
Higuain arrived with a prolific record to suggest that he was an ideal candidate for the job, but has managed just three goals in 11 appearances.
Questions have been asked of his suitability to life in the Premier League, with Sarri among those to have expected more, but the South American is eager to seal a full-time deal this summer.
“It’s always nice to try a new league and it’s my first taste of the Premier League.
“Obviously it’s a period where I’m adapting but I’m hoping that, although I’ve only got a short time, I can finish well. The best thing we can do is qualify for the Champions League and hopefully win the Europa League too.
“We’re coming towards the end so it’s about finishing as strongly as possible.
“The main objective is to finish in the top four so we can play in the Champions League and also to encourage other players to come and play for the club. That’s what the squad are hoping for, so that we can add players next season, and to finish the season well and make people happy.”
He added: “From a personal point of view I want to do as well as I can so I can be here for next season. That’s my intention.
“I came here and I wanted to stay. It’s a good city, you live well and you can be relaxed but also enjoy yourself. So my intention is to stay here.”
He said: “You watch the games on TV and you see that it’s a fast and powerful league to play in, but when you’re playing it’s even more so. You come up against defenders that are really strong.
“Also, we’ve seen any team can beat any other team. We’re one point off the Champions League spot so we have an opportunity to put pressure on the other teams for the end of the season.”
Chelsea can move into the Premier League’s top four on Monday by taking a positive result from a derby date with West Ham.
After that contest, their attention will then shift to the first leg of a Europa League quarter-final clash with Slavia Prague.