The Scot has had little to celebrate this week following last Saturday's shock FA Cup exit to Wigan and a selection of fans voiced their displeasure following the quarter-final defeat.
There has since been more speculation over his future at Goodison Park, with his contract due to expire at the end of the current campaign.
However, reflecting on his time in charge in his programme notes ahead of their Premier League win over Manchester City, Moyes appeared to suggest he has unfinished business with the Toffees.
He wrote: "I believe so much has changed since that very first game.
"The progress that the club has made since then can by shown by Everton's Premier League standings, when in the last 10 seasons we've finished outside the top 10 only twice.
"I understand this will never satisfy everybody and nothing would give me more pleasure than to bring the success that Everton had in the 1980s under Howard Kendall, with the great players that played for him.
"We are striving to reach those levels and we put every ounce of effort into trying to get back to that standard."