Arsenal have had bids with both £15million and £25m rejected so far this summer with Celtic unhappy with how long it would take them to received the latter sum in full.
Unai Emery is determined to strengthen his defence with the signing of the Scotland international, but are yet to make an improved bid.
The news was confirmed by Hoops boss Neil Lennon on Thursday, as he preparing to start the new season with his star man in the squad.
“No, not that I am aware of,” the Northern Irishman said when asked if a third bid had been made.
Gunners fans were left excited on Wednesday when Emery promised that the club were in the market to improve their playing squad before the window closes in three weeks.
He said: ”Our target is to achieve three, four players to really improve our team and squad now.
“We are being very demanding and we are also first speaking about the possibility to sign very big players, very expensive.
“The first player in our list is the first target and after, if we cannot have that then [we try to sign] the second, but thinking every time that they are going to improve our team.”
Arsenal missed out on Champions League qualification last season with a fifth-placed finish in the Premier League, and this summer’s transfer budget was then reduced.
Club hero Charlie Nicholas has supported a statement from fans that has criticised Arsenal’s ownership, and urged them to make changes.
“I totally support the Arsenal fans’ statement,” he told the Sky Sports Transfer Talk Podcast.
“The concern has been growing for 18 months, about the lack of atmosphere, lack of ambition. I think the criticism is fair.
“Now there’s a situation whereby there are too many chiefs trying to have a say. Are there too many guys sitting on the committee? It’s getting a bit out of control.
“If Arsenal are going after defenders, which they need more than anything else, then go and nail them down, go and make a statement of intent.
“Arsenal have to force the issue. I know times have changed, but Arsenal are finding it damned hard to compete at a big financial level.
“It doesn’t seem as if there is a strategy, and that is a true concern.”