Allardyce is under contract for another 12 months but Moyes only until midnight after the final game of the season, meaning there is no certainty he will still be in charge at the London Stadium next season.
The Scot, appointed in November as successor to Slaven Bilic, steered West Ham to Premier League safety in a campaign capped by two goals from Manuel Lanzini and Marko Arnautovic’s 11th of the league season.
Moyes said: “Everybody always thinks the grass is greener and there’s something better out there. I would say more often than not I bet you it’s proved wrong.
“We’ve shown we’re trying to improve. I don’t know if anybody can come and make that big a difference in six months. Even (Manchester City boss) Pep (Guardiola) took a bit of time to get his team right.
“I’d think you’d need to give a bit longer. But every manager now is in transit. I don’t think there is a long period (to settle in).”
Moyes may be persuaded by the potential of his team, demonstrated on Sunday.
We end the @premierleague season in 13th place.
Thank you for your incredible support this year! pic.twitter.com/uEzOm0GQqI
— West Ham United (@WestHamUtd) May 13, 2018
“What will influence me? When you get good performances,” he added.
“It makes you feel ‘hey, there’s something to build on here’.”
Asked if he was now officially out of contract, Moyes said: “Midnight. I’ll take my suit off at midnight, undo my tie a bit…”
Moyes expects to meet the West Ham board, co-chairmen David Gold and David Sullivan and vice-chairman Karren Brady, next week to determine his future.
He spent 11 years as Everton boss and was asked his thoughts on the negative reaction of Toffees fans to Allardyce.
He added: “Am I surprised? Yes, I am a little bit. They’ve always been a club who get behind their players and their manager.”
Allardyce gave a curt response to questions on his future.
The former England and West Ham boss said: “Contract for next year. Confident? I can’t quite say confident after all the rumours I’m reading in the paper.
“Generally there’s no smoke without fire. We’ll wait and see when we meet up.”
Allardyce was also asked about the future of Everton’s Wayne Rooney, who was absent here with a knee injury and has been linked with Major League Soccer’s DC United.
Allardyce added: “It’s up to Wayne and his representatives and the board of directors and what they do. Everybody will know by the end of the week.
“And just to stop everybody talking and writing nonsense: there’s no problem between me and Wayne Rooney. We’ve been very professional and very adult with our conversations.”
Allardyce felt Everton had the better opportunities, only to be frustrated by West Ham goalkeeper Adrian.
Moyes feels the Hammers have been transformed since his arrival.
He said: “When I go back to when I took over, and how I was trying to get results by hook or by crook, today it certainly looks much more structured and we’ve got a way of playing.
“They were very good today, the players.”
Chapman led Huddersfield to two successive league titles in the 1920s before repeating the feat at Arsenal, who went on to dominate English football in the 1930s.
The 68-year-old Frenchman had taken a quiet moment to himself in the pitch-side dug-out an hour before kick-off and took time to reflect.
“I wanted to get a bit of oxygen, my last day, nice weather, people were nice, Huddersfield were happy, they gave me a nice present,” he said after his side had won for the first time on the road in the league this year.
“Herbert Chapman came from here. For me to come here on the last day was a special meaning when you know the history of our club.
“Chapman smiled at me as he was on the photo just outside the dressing room.”
Chapman was tempted into leaving Huddersfield to take charge of Arsenal in 1925 after setting the West Yorkshire club on their way to a historic hat-trick of league titles.
— Arsenal FC (@Arsenal) May 13, 2018
But while Wenger revealed he had already had offers from rival Premier League clubs, it was unlikely he would take another job in England so as not to face Arsenal.
“I’m not ready for that at the moment,” he said. “That would be very difficult, I’d stay at home that day. Maybe I don’t envisage that at the moment.
“Yes I had offers, yes, of course. I had many offers. But at the moment, I have not said anything to anybody.
“I came out of a long process and you cannot just the next morning go somewhere else. It’s impossible.”
Asked if he will take another job in England, he said: “I don’t know. Maybe it’s better I go somewhere else.”
Wenger won three Premier League titles and seven FA Cups, including the double in 1998 and 2002 and an unbeaten league season in 2003-04.
He had become a divisive figure at the Emirates Stadium but he felt the fans’ love in his final match.
“I feel that I got a lot of respect not only from our fans, but from England,” he said. “I would reiterate I loved English football, but I also learned to love England.
“You do not stay 22 years if you don’t like it. That’s a good return from people who love football here. It’s a special country, especially for football and that’s why I stayed a long time.”
— Arsenal FC (@Arsenal) May 13, 2018
Huddersfield boss David Wagner, who made a special presentation to Wenger in the players’ tunnel before kick-off, had spoken earlier in the week of his admiration for him.
Wagner admitted his players’ job had already been done after Wednesday’s draw at Chelsea had guaranteed their improbable survival.
Terriers fans’ celebrations made for a carnival atmosphere in the sun and Wagner was delighted his club had played a part in Wenger’s send-off.
“It was great we were able to show him all the honour and respect that we have. We were very happy to be a part of his special day,” Wagner said.
Speculation continues over Wagner’s own future. He will sit down with club owner Dean Hoyle next week to discuss the future.
When quizzed about Hoyle telling fans that the manager would be going nowhere in the summer, Wagner joked: “Well he’s put himself under a lot of pressure hasn’t he?”
City assistant coach Arteta, a former Gunners player, is reported to be one of the prime candidates to take over at the Emirates Stadium following the end of Arsene Wenger’s 22-year reign.
The 36-year-old began working with City after retiring from playing in 2016 and performed a key role in this season’s title success.
Guardiola, speaking after champions City beat Southampton to become the first club to accumulate 100 Premier League points, said he “wants the best” for his friend and colleague Arteta.
“What we have done this season, Mikel his contribution was outstanding, amazing, we work together so good,” said City boss Guardiola.
“If he stays I will be the happiest guy in the world and if he decides to move because he has this offer, this option, I will not say you do not have to go.
“I want the best for my friends and he’s a friend of mine.
“If he decides to go, I will be so sad but I will understand his decision because it’s his career, his life, his family, and I am not the right guy to say you don’t have to do that.
“But hopefully he can stay and finish what we started together in the upcoming years.”
While City ended the season with a 1-0 win at St Mary’s, Gunners boss Wenger signed off by beating Huddersfield by the same scoreline.
Juventus coach Massimiliano Allegri, former Chelsea boss Carlo Ancelotti and ex-Gunners captain Patrick Vieira are among the other names linked with replacing the long-serving Frenchman.