The Toffees currently sit eighth, six points behind fifth place Everton.
"I think Everton should be kicking around those places, I really do," said Moyes. "All the years they were not were wasted years. Everton should be attempting to be (in Europe).
"We might not be good enough to be it, we might be short, people might write something (later in the season) because I was talking about it but I have been trying to make Everton more positive.
"A better look, a better feel, better results all in the hope that… hey, you never know. It might be too soon this year. We could lose two games and I’ll be saying it’s great we have avoided relegation!
"But at the moment, the position we are in, we have to keep thinking we can challenge the top. I am trying to drive that through and want the players to hear it. That’s the plan.
"Fifteen years ago (in Moyes’s first spell) it was terrible, we could not break the ceiling of the top teams. We were trying to break it, be better than those four or five teams. Every year, getting to the Champions League felt impossible just about.
"We did it once (in 2005) so it wasn’t. But now, the competition in the mid-band of teams, someone in 15th could maybe still get to Europe if they go on a run. So if we can just keep nudging the points along… we’ve had a couple of disappointing draws but they’ve helped."
Transforming
Moyes won the Europa Conference League with West Ham and he continued: "Looking back I think now, 'My goodness, those years in Europe made some difference to West Ham (his former club) as a football club'.
"Some difference to the finances, the way people see it. We had a semi-final, a quarter-final and final in three years.
"Next weekend, because we are knocked out of the FA Cup, I’ll be sitting twiddling my thumbs so I want us to be involved. I am too old to sit here and say I’m just going to try and keep us away from the bottom of the league."