Leno made 32 Premier League appearances for The Gunners last term after joining from Bayer Leverkusen for £19million.
He kept seven top-flight clean sheets but played only four times in other competitions.
Emery anointed Petr Cech as back-up goalkeeper, playing him in cup games and, notably, the Europa League final, even though it was reported the Czech Republican would join fellow finalists Chelsea as a director after retiring.
Now Cech has departed, Leno wants to nail down a starting spot in every competition, according to the Evening Standard.
Academy graduate Emiliano Martinez, who has featured just six times for Arsenal in seven years, is expected to be promoted to the bench.
Arsenal have not ruled out signing another goalkeeper and would ideally add a more experienced shot-stopper against whom Leno could challenge for a starting spot.
But other areas of the side are in greater need of improvement and, with £40m available in the market, the club cannot be frivolous.
Whether Leno’s wish will be met is unknown, with Martinez, who was on loan at Reading for the backend of last season, likely to feature in EFL Cup games.
The Argentinian is not at the level of Cech, though, which would make Emery hesitant about playing him in the Europa League – potentially the club’s best route to the Champions League again.
If Arsenal are to spend above £40m this summer that will have to come self-funded player sales, with Shkodran Mustafi transfer listed by Emery.
On the matter, Gunners insider David Ornstein told BBC Radio 5 Live Sport: “Arsenal, as we know, also want a central defender. It’s emerged that William Saliba from Saint-Etienne is one of their options.
“The suggestion is he’d be loaned back for a season which wouldn’t be ideal as they need a centre-half now. There’s no confirmation on that but talks are ongoing.
“I was told that after the Europa League final that Unai Emery, who had always put his faith in Shkodran Mustafi despite a number of high-profile mistakes, has decided enough is enough with him and he is no longer in favour of keeping him, so he could be sold.
“Arsenal have tried to sell him almost every window since they signed him.”
Arsenal also want to sign Celtic left-back Kieran Tierney, for whom they had a £15m bid rejected earlier this window.
