The Toffees have made a bright start to the season and are pushing for a European spot, with confidence high around Goodison Park.
Roberto Martinez's side have already taken one notable scalp, after comfortably dispatching of Chelsea, and Naismith felt that result highlighted the quality that they have in their squad.
"The top four has now probably turned into a big six or seven," he said. "Spurs are consistently fighting for it, we did it a couple of years ago. We feel we're good enough to be up at that mark.
"We're confident against anyone. If we play as we know we can, then we are a match for anyone in the league. We have shown that over the last two or three years."
Everton are under pressure to have a productive season this year, as the last campaign was a disappointing one for the Merseyside club.
The Toffees could only muster a mid-table finish in 2014/15, but there are clear signs that they have made improvements over the summer and are looking a threat again.
Martinez's side have been solid in defence and explosive going forward, and they currently sit in seventh place in the Premier League table.
The international break rather halts the momentum that Martinez's side have built up, but when they return to action, Everton will have a chance to prove Naismith's claim correct.
Everton take on Manchester United and Arsenal in their next two fixtures, and it provides them with a good opportunity to make a statement of intent for the season ahead if they are able to get some points from those matches.