To do that, they will have to overcome opponents who have risen to third in the table on the back of a superb run of results just as the race for Champions League qualification enters the finishing straight.
With Manchester United and Manchester City set to finish first and second, Arsenal, Chelsea and Tottenham Hotspur have been jostling for the remaining two Champions League spots for the last three months.
Yet victory at Arsenal will make Everton serious contenders too.
United, who need just seven points from their last six matches to win the title lead the pack with 80 points from 32 games, 15 points clear of City who have 65 from 31.
Arsenal have 59 points from 32, followed by Chelsea on 58 from 31, Spurs on 58 from 32 and Everton 55 from 32.
Tuesday's match at Emirates Stadium is one of four important midweek Premier League games before a full programme next weekend.
Runaway leaders Manchester United play at West Ham United on Wednesday, fourth-placed Chelsea play at neighbours Fulham and Manchester City host relegation-threatened Wigan Athletic in an FA Cup final rehearsal.
Arsenal, who are chasing a 16th successive Champions League berth next season, have recovered superbly since they were beaten 2-1 at arch-rivals Spurs on March 3.
They have won all four league games since then, including Saturday's 3-1 victory over Norwich City when all three goals came in the last six minutes.
Everton's European ambitions have taken root with four wins and a draw from their last five games, with coach David Moyes praising his team's professionalism in their 2-0 win against relegation candidates Queens Park Rangers on Saturday.
"It was a very important win and a very disciplined and professional performance. Now, if we can go and win at Arsenal, you might say that we've got a chance of being involved in the Champions League.
"If we don't, you'll probably say we've got a chance of making the Europa League. No matter what happens there, the players have given themselves a great chance."
He added: "There's six games left to go. We've been doing it for most of the season. Why would it change in the last few games?"
What Everton need to change is their record against Arsenal. They have lost eight of their last 11 league matches and not beaten them for six years.
Manchester United, who brushed aside a toothless Stoke City 2-0 on Sunday, can take another step towards their 20th title if they win at mid-table West Ham who drew 1-1 with Southampton on Saturday.
A last-minute goal from Robin van Persie gave United a 2-2 draw at Upton Park on their last visit there for an FA Cup third-round tie in January.
They have a good record at the ground, winning their last three league matches there and outscoring the Hammers 9-2.
In a dress-rehearsal of the FA Cup final on May 11, Manchester City meet Wigan at home with the game of far greater significance to their opponents.
Wigan are 18th and in the relegation zone, but have two matches in hand on the teams above them.
Chelsea, who relinquished their hold on the FA Cup on Sunday when they lost 2-1 in their semi-final against Manchester City, will be playing their seventh match in 19 days, when they make the short trip to Fulham.
Fulham's Craven Cottage ground is just two miles from Chelsea's Stamford Bridge.
Chelsea usually do well at Fulham having lost there only once in the Premier League, and interim coach Rafa Benitez will be desperate for all three points to keep their momentum going in the race for Champions League qualification.
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