Arsène Wenger insists his managerial skills remain as sharp as they were during the Invincibles' season, despite the grumbles of Arsenal fans unhappy with their team's diffident start to this year's campaign.

Wenger, whose decision to substitute Olivier Giroud for Francis Coquelin during Saturday's 0-0 draw at Aston Villa was met with boos and cries of "You don't know what you're doing", believes those "extreme opinions" don't reflect the views of most supporters. But ahead of Wednesday's trip to face Everton at Goodison Park he has urged fans to be realistic about his side's capabilities.

"Last season we finished third," he said. "Honestly I don't think there was much more in that team than third. Like when we won the Premier League with zero defeats, which never anybody else has done, you think there is not much room to do much better. At the end of the season I want to stand in front of my mirror and think: 'Have I done all I can?' That's all."

Wenger remains unworried by the persistent brickbats from some fans. "In the modern media, extreme opinions are picked and sold like a majority, with those who shout the loudest usually presented because they are interesting and easy to sell," he said. "We don't give too much importance to those fans' reaction. Our job is somewhere else, on the football pitch."

Everton, who lie fifth – a place and a point ahead of Arsenal – are an important staging post in the Gunners' season. After Wednesday's match Wenger's side have home games against Swansea and West Brom, trips to Reading and Wigan, and then further home games against West Ham and Newcastle. There are more daunting schedules, and it is not unreasonable to expect that by the new year the mood at the Emirates could be much rosier.

But an Everton side with an imposing home record – they have not lost at Goodison since Arsenal beat them in March – will have something to say about that. "Every big team in the league goes there thinking they're in for tough game," said Wenger, who equated their manager David Moyes's achievements on a limited budget with his own: "Of course I admire him but I can show you our balance over the last 16 years and you would be astonished."

Arsenal's new £150m sponsorship deal with Emirates has provided optimism that Wenger will spend in the January transfer window, but he remains wary of rivals with deeper purses. "We are now going towards a period where we will be able to compete again financially with other clubs," he said. "And we will be on the market if we find a top player. But Paris St-Germain have unlimited resources. If they decide to go to £100m, they do it. If they decide to go to £150m, they do it. They have a building called the Shard that cost £1bn, that was built by Qatar. I cannot promise we will match them."

It was telling that when Wenger was asked whether the new sponsorship was more about helping him not losing players, he replied "Yes, of course."

Wenger is hopeful that talks with Bacary Sagna over a contract extension will reach an amiable conclusion but there was no fresh news over tying Theo Walcott to a long-term deal. Walcott returns for Wednesday's match, but so does Everton's Marouane Fellaini – a player Wenger greatly admires. "Their offensive game turns around him," he said. "He is the target man they look for. He is not a dirty player. He is an intelligent player."

Could he do a job for Arsenal? A mischievous smile, and a single word. "Yes."