The Gunners clinched fourth place and guaranteed Champions League football for a 16th consecutive year after securing victory over Newcastle on the final day of the Premier League, but there has been growing discontent amongst fans following eight years without silverware in north London.
Club legend Adams, who spent 22 years at the club and captained the side to 10 major honours, said: "The best players and the best teams win championships.
"Lately Arsenal have become very solid and (Laurent) Koscielny and (Per) Mertesacker have gained a confidence together, but defensively I think they are still short - I'd like to see them sign another centre-back, a couple of full-backs and possibly a keeper."
The former England captain, who won three league titles in three different decades at the club, admitted that he understands the frustrations of the fans, but urged them to be patient with the young side.
"The fans are desperate for a trophy of some sort," he added. "I would like the players to get used to winning; if you have never had that experience of crossing the line as a winner you don't know what it is like to be a winner.
"Jack Wilshere is very important to the club, but you can't rush him. I think he has played more games than Arsene (Wenger) would have wanted him to play following his injury, but he's such an influential player for us."
Adams has no doubt that Wenger is still the man for the job - a sentiment reinforced by World Cup winner Pires, who spent six successful years under his fellow Frenchman at Arsenal, where he won two Premier League titles, and three FA Cup crowns.
"Wenger built Arsenal so he is very important to the players and needs to stay," Pires said.
A prominent member of the renowned 'Invincibles' side, Pires played alongside the likes of Dennis Bergkamp, Thierry Henry, Sol Campbell, Patrick Vieira and Ashley Cole as the club went 49 games unbeaten including the entire 2003-2004 season - and he has urged his former boss to look to the British market when targeting possible signings this summer.
Pires added: "In my opinion I think Arsenal need to buy English players.
"He (Wenger) has changed the philosophy at the club, he brought in young players like (Theo) Walcott, (Alex-Oxlade) Chamberlain and (Aaron) Ramsey, these players are very good for the future but now they need to buy some players with experience to give a good mix."
Tony Adams and Robert Pires were talking to Sky Sports after taking part in the Indesit Support your Hero event.
Thanks to Indesit, fans were given the incredible opportunity to be trained by legends from AC Milan, Arsenal and Paris Saint-Germain in the San Siro Stadium as part of the trip of a lifetime.
People from more than 30 countries registered for the contest through the Indesit website.