Manchester City officially announced the €60 million signing of Haaland from Borussia Dortmund on Monday, while Liverpool have agreed a deal to sign Nunez from Benfica, the Portuguese club have announced.
Kane had wanted to leave Spurs last summer to join eventual Premier League champions City, but Spurs chairman Daniel Levy rebuffed all interest from Pep Guardiola's side, who opted to not to match Spurs' £150m (€175m) valuation.
The 28-year-old is now expected to remain in north London as head coach Antonio Conte aims to turn Spurs into Premier League title challengers.
Kane has won the Golden Boot as the division's top goalscorer in three of the past seven years, but last season he finished behind his Spurs teammate Heung-Min Son, who tied with Liverpool's Mohamed Salah for the award with 23 goals.
Asked on Monday whether Haaland and Nunez arriving in England would make it harder to lead the goalscoring charts, Kane told a news conference: "The battle for the Golden Boot is always tough. The Premier League has produced some top strikers from around the world for a number of years now.
"You expect the top strikers to want to play in the Premier League and with those two new signings, that is going to be the case. It helps. It helps me as a player to have good competition, it drives me to improve and get better. So for sure, I look forward to the challenge.
"I try and focus on myself to begin with. Going into any season, I have things I want to achieve and goals I want to reach. I try not to focus too much on other players in that aspect.
"I can't control what they do but what I'll do is continue to work hard, improve and after this game have a nice break. But then look forward to what will be a tough preseason, I'm sure, and get ready for the new season as I have done for the last seven, eight years now."
Kane is on international duty with England, who face Hungary on Tuesday in the last of four Nations League games this month. He is now three goals shy of Wayne Rooney's all-time international record of 53.