Arsenal revealed on Thursday that captain Koscielny had refused to join up with his team-mates for their three-game tour pre-season tour of the United States.
“My captain, my a**e," Groves said on talkSPORT: “I'll just read out a quote: 'I don't believe any French player would have refused to come and play for Arsenal. It took me four years to go from the bench of Ligue 2 to Arsenal, so it is a great achievement. I have been on the crest of a wave for the last few years but it is just the beginning, I haven't proved anything yet. Arsenal is a reference for football.'
“Who said that? Laurent Koscielny when he signed! I think someone should have a word with him and say, 'Laurent I think you've forgotten your beginnings, I think you've forgotten when you've come from'.
“I hate it when footballers refuse to play, because they forget the privileged position they're in.
“They're in a very small minority of people who love what you're doing and get paid, he's been paid handsomely for nine years at Arsenal whether he's been injured or out of form, they stuck by him.
"If I was a younger player who was being advised and had an agent saying you've got to force a move… but he's 33 and he's made the decision on his own!
“For him to try and force a move away from Arsenal is an absolute disgrace. They should fine him and strip him of the captaincy.
“It would be great to say you're never going to play for the club again, but Arsenal are not in that position because they haven't got enough players.
“And thing he's done as well is, he's been there nine years and you wouldn't put him in the Frank McLintock, Tony Adams, Steve Bould, Sol Campbell, Kolo Toure class, but he's been a good player for us and he's won two FA Cups.
“But, he'll have no affection or good thoughts from the Arsenal fans at all now."