Despite added competition from Chelsea and Arsenal, the 24-year-old is adamant that the Merseysiders are still in with a shot of a high finish.
"We need to aim as high as we can," Anichebe told Sky Sports News. "We just need to keep plugging away, win as many games as we can, and we'll be in and around it. We're not far off now. There's definitely a chance.
"We were disappointed we didn't win. It was a really good performance against Spurs. It says a lot about our desire because we're missing two big players.
"Whoever comes in does well - and that just shows what the manager has instilled in us. When you get your chance, you have to take it."
Everton host Queens Park Rangers in their next match this Saturday.