Under David Moyes and his limited budgets, Everton have maintained a solid record in the Premier League. Consistent in their contention for European places, Everton have regularly won against Big Four teams under his tutelage. He's been manager since 2004 – to give you an idea of how long that is, that was the Season of Arsenal's Invincibles – and though his results, purchases and reputation are nearly beyond reproach, he's not been heard of in reference to either Big Four jobs available during his Toffee reign.
Occasionally touted as the successor to Sir Alex Ferguson at Manchester United, as much for his Scots blood and ability to throw the odd tanty as for his results, Moyes has worked well as manager of a second level club and has probably overachieved. That consideratiion for big jobs hasn't come straight away because those “big” jobs have only come up at Chelsea and Liverpool. Chelsea under Abramovich wants glamour, while the move to Liverpool from Everton is about as likely as Moyes' next job being the Prime Minister of Kenya.
In Sir Alex Ferguson or Arsene Wenger were to call it a day, would he get consideration? Given his understated demeanour there may be fears he's too low-key for such a big job. Low key isn't necessarily bad, it's just a different modus operandi than you would expect at a United or and Arsenal. Other muttered concerns include him “finding his level” in that second level with Harry Redknapp and Martin O'Neill and whether he has the requisite European experience to succeed at a big club.
Of all the EPL managers, Moyes is probably the most deserving of that big break. But this doesn't mean he will automatically receive it – there's no waiting your turn in football. One bad season could spoil eight years of hard work and good results. One bad season could mean his next job is in the relegation spots, the Championship or, heaven forbid, the SPL. The example of David O'Leary still hits home hard: linked with Premier League jobs as recently as Portsmouth last year but now finds himself managing in the United Arab Emirates simply to stay in work.
Perhaps one day his talent will be recognised for what it is but the guess is here that he will be at Everton for a good few years yet.