The Toffees boss would take a spot in Europe's second-tier competition after a draw with Arsenal looks to have removed his side from the race for Champions League qualification
The Toffees drew 0-0 with top four rivals Arsenal at the Emirates Stadium on Tuesday night.
The result left Everton two points behind fourth-placed Chelsea - who have two games in hand on Moyes' side - and reduced their chances of making the top four.
|15/2||Everton are 15/2 with Coral to finish in the top four this season
"Of course we can do it but I’m a realist. We’re going to try and get Champions League - if we’re talking about that in mid-April as Everton then we’re not doing bad.
"We have a small chance but are we going to overhaul three teams? I don’t sense vulnerability in Arsenal. It will suit Spurs more than anyone that result tonight.
"We’re not out of place where we are, it’s a different playing field for us but we’re going out on the pitch doing everything we can.
"A point keeps us in with a chance of Europa League which we must try and do.”
The Scotsman was content with the point earned, saying his side deserved the draw but not much more.
He continued: "I’m pleased with the performance and can’t be disappointed with a draw at the Emirates.
"We had bits of defending that were terrific and we more than thoroughly deserved a point. We didn’t do enough to win, I was looking for more chances and opportunities than we created.”
One of the standout performers was Ross Barkley, and Moyes is keen to continue to give the youngster, and some other youth players, their shot in the first team.
He added: "I want to give the young boys a chance. It’s hard to do that especially when you have something to play for.
"We are having to find the right time to play Ross. He’s got a bit to do but a lot has been talked about him and we have to bring him on slowly. He’s not ready to be a regular.
"I wasn’t nervous about [Darron] Gibson [getting sent off] because we made it a right good game; we got the crowd going because were committed. Not everything has to be a foul.
"Were we rough, sometimes that’s allowed. We are not going to come here and let them stroke the ball around and make 500-600 passes."