The former Preston North End boss will celebrate 11 years in charge of the Merseyside team next month with a top-four finish and an FA Cup run possibilities this year.
The Blues have enjoyed a successful league campaign so far this season and are currently sixth in the table. They will face a home clash with Wigan Athletic in the FA Cup quarterfinals if they can see off League One side Oldham Athletic in a fifth-round replay next week.
But is Moyes right to leave any decision on his new contract until the end of the campaign?
When the former Celtic defender arrived at Goodison Park to succeed the sacked Walter Smith, Everton were facing a potential relegation battle and had just been knocked out of the FA Cup in a demoralising defeat at Middlesbrough.
Moyes was considered one of the best of the young breed of managers in the Football League after taking Preston to the 2001 Championship playoff final where only defeat to Bolton Wanderers prevented him leading the Deepdale club into the top flight.
The call of Everton was strong, though, and he was introduced to the supporters ahead of the Premier League game against Fulham on March 16, 2002, which the Blues won 2-1 to immediately ease fears of the drop.
What Moyes has done at a club that had significantly underachieved in the preceding years, including two last-day survivals in 1994 and 1998 is quite astonishing considering the Merseyside club have not enjoyed the financial wherewithal of others.
Moyes has been rewarded for his efforts with three LMA Manager of the Year awards in 2003, 2005 and 2009, but the lack of a trophy will hurt the Scot.
He will be looking enviously toward the Capital One Cup Final between Swansea City and Bradford City this weekend, after Everton were knocked out in a woeful performance by Leeds United earlier in the season.
The closest Moyes has got in his time at the club was the 2009 FA Cup Final where even Louis Saha's quickest-ever final goal was not enough to see off an in-form Chelsea under interim boss Guus Hiddink.
And his no-nonsense approach to high-profile players at clubs will also be an attraction for Chelsea owner Roman Abramovich.
Moyes banished Everton striker Duncan Ferguson from the training ground in November 2003 after a disagreement between the two men as the official Everton website recorded in an overview of the season.
However, after enjoying a stable relationship with Everton chairman Bill Kenwright since the theatre impresario officially took up his post in 2004, would Moyes want to walk into the turbulence which can surround Chelsea, where nine managers have worked under Abramovich since he took over the club in 2003?
Manchester City have also emerged as a possible destination for Moyes if Roberto Mancini exits the club in the summer, but the Scot has managerial etiquette of not speaking to or about other clubs with an incumbent still in situ.
However, Moyes' long-term strategy displayed at Everton will be a keen advantage in the eyes of City owner Sheikh Mansour and his Abu Dhabi United Group, who are keen to ensure the club remains a mainstay of success at first-team and grassroots level.
But Ferguson remains an avid admirer of his fellow Glaswegian, and Moyes almost became his assistant in 1999, as he told the Daily Mail.
Moyes voiced an interest in a move to Germany in an interview with France Football, which was reported by Sky Sports, and both Schalke and Werder Bremen have been credited with an interest by the Daily Star.
Should David Moyes leave Everton at the end of this season?
However, the question remains whether Moyes should be jumping the Goodison ship? His loyalty toward the Merseyside club has been highly commendable, and he deserves his place in Everton's Hall of Fame.
But his most recent comments about his future appear to align with a man "fishing" for offers rather than a design to maintain his progress at Goodison Park.
Uncertainty over a manager's future can unsettle players within a squad but can also act as a catalyst for improvement, but Moyes' statement does not seem designed for the latter.
Moyes has proved himself time and again with Everton, especially in recent seasons, but the collapse of the deal to bring midfielder Leroy Fer to the club in January was a significant blow to his plans towards of a top-four finish this season.
The Everton boss had previously claimed he would discuss his future after the January transfer window, as Goal.com reported, but The Guardian recounted his admission that failed transfer bids could hurt his hopes of claiming a Champions League berth.
Delaying contract negotiations further than Moyes originally intended does not bode well for Everton supporters wanting to see the manager remain in place.