The France international forward is among those expected to be moved on by Jose Mourinho when the next transfer window opens.
Emirates Stadium has been mooted as one possible landing spot for the 22-year-old, as a proven Premier League performer.
Groves, though, believes that Arsenal should setting their sights a little higher in the recruitment department than a player deemed surplus to requirements at a direct rival.
The Gunners cult hero told Sportingbet: “Anthony Martial is a quality player but I don’t want Arsenal to sign him this summer.
“Martial isn’t a priority for us because we have enough players going forward in those areas.
“He isn’t deemed good enough for Manchester United so I don’t want him at Arsenal.
“Arsenal’s priority needs to be a new goalkeeper, a new central defender and a very strong defensive midfielder, but that’s been the case for five or six years now.”
The job of identifying and then landing any transfer targets will be passed on this summer, with Arsene Wenger preparing to step down as Arsenal manager after 22 years at the helm.
Groves said of that decision: "This summer is right time for Arsene Wenger to leave Arsenal. It’s the right time for the club and it’s the right time for Arsene.
“He’s been receiving abuse from a section of the Arsenal fans that has been far too strong and it was affecting him. I think it also became apparent that Stan Kroenke’s support wasn’t as strong as it once was and Arsene felt the time was right for him to leave before he was eventually pushed this summer."
The Gunners are now in the process of appointing a successor to an iconic figure, with Groves of the opinion that they should be looking to the continent for experience at the very highest level.
Pressed on who he would hand the reins to, the former midfielder said: “Luis Enrique is my first choice to replace Arsene Wenger. Massimiliano Allegri is second and Diego Simeone is third.
“Arsenal need a manager who has that winning mentality, who has managed a big club, has proven he can manage big players with big egos and can instil a similar philosophy similar to what Arsene has instilled over the last 22 years.
“Arsenal need to appoint a big-name manager who can attract world class signings, particularly if the club doesn’t have Champions League football next year. Luis Enrique fits the bill perfectly, he has a massive pedigree and he speaks perfect English.”