Roberto Martinez's side is hoping to leapfrog Arsenal this weekend, while Gus Poyet is aiming for a "miracle" to beat the drop into the Championship this season.
Gus Poyet's side is rooted to the foot of the table, seven points from safety, after picking up just one point from its last seven top-flight matches.
The club's next three fixtures would appear to offer little room for optimism, with games at title contenders Manchester City and Chelsea following the visit of Everton.
By the time fellow straggler Cardiff City arrives at the Stadium of Light on April 27, the situation could be beyond repair - something Poyet acknowledged following Monday night's 5-1 humbling at Tottenham.
"I think we need a miracle. We need something unique, a shock, because if not I cannot see it coming," said the Uruguayan.
By contrast, Roberto Martinez's Everton is very much on the crest of a wave following last Sunday's stirring 3-0 win over Arsenal at Goodison Park, a result that leaves the club in control of its destiny when it comes to securing qualification for the Champions League.
On the same weekend Sunderland fell foul of City at Wembley, Everton beat West Ham 1-0 to launch a winning run that presently stands at six Premier League wins.
Belgium international Romelu Lukaku came off the bench to net a late winner against West Ham following an injury lay-off, and the influence of the on-loan Chelsea striker on Everton's subsequent charge cannot be underestimated.
Deployed unconventionally on the right side of Everton's attack last time out, he unsettled Arsenal's back line throughout and stormed in from the flank to score his 14th goal in 27 appearances for the Merseyside club.
A point behind fourth-placed Arsenal with a game in hand, Martinez's men are now emerging from their status as fancied outsiders to be seen as a team expected to succeed.
However, Everton will not be taking Sunderland lightly this weekend, having been beaten 1-0 in December's reverse fixture.
"They've had significant performances at home this season and they can replicate that level at any time," said Martinez of Sunderland on Thursday. "We'll have to be at our best against a team that's ready to fight for every ball and you have to have the same level of focus and concentration.
"We remember the defeat at home against them and we need to be as good as we can be."
Martinez provided an update on midfielder Leon Osman, who was forced off early against Arsenal with a facial injury.
He added: "Leon Osman is the only doubt we have. The cut is healing well but the eye is not fully open so we need to make sure that doesn't affect his vision."