New boss Carlo Ancelotti has reinvigorated Everton with 17 points from eight matches - bettered only by near-neighbours and runaway Premier League leaders Liverpool (24) over the same period - catapulting them to within five points of the top four.
"We're trying to catch those European places. A few months ago, we probably didn't think that was possible," Coleman told evertontv.
"But we're playing well now and just looking forward to every game.
"We've got to push for Europe. I think this club needs European football and we all want that.
"Where this football club wants to get in the next three, four, five years, it has to be wanting European football every year and if we can get it this year, that's a bonus."