The Gunners are weighing up a deadline day bid for Besiktas defender after the Turkish club placed a £26.9m price tag on the World Cup finalist.
According to the London Evening Standard, Arsenal wanted to do a cash-plus-players deal which would see Danny Welbeck or David Ospina go the other way, but Besiktas would prefer a straight sale.
Davis told Sky Sports News (quoted by the Daily Express) that Arsenal's defensive problems could force them to invest in a centre back.
"If it is to be anybody, it’s likely to be a defender," she said. "
"Lots of speculation around the Croatian defender over the past few days, Domagoj Vida.
"The president [of Besiktas] has made it quite clear he was in London, he was Instagramming pictures of himself in London Park yesterday.
"And he has been quoted in saying ‘If it’s in the best interests of the club, we will let him go’.
With the Gunners having already spent over £70m this summer, it is understood that any more purchases will be dependent on player sales.
To this end, Calum Chambers joined Fulham on Tuesday and Arsenal are also attempting to shift Lucas Perez, who is closing in on a move to West Ham. The sale of Chambers could hint at a new centre back arriving before the deadline.
Vida has also been linked with a move to Liverpool, but Jurgen Klopp looks set to keep faith with Vida's international teammate Dejan Lovren, having effectively ruled out the possibility of any deadline day arrivals at Anfield.
Vida. 29. Signed by Besiktas in Jan 2018 on a free. @Arsenal are considering spending £25m on him because he had a decent World Cup. Post-Wenger this club is still terrible in the transfer market. Torreira the only unquestionable piece of business— King (@KingLafs) August 6, 2018
With interest also reported from Italy and Spain, Besiktas are in a strong negotiating position, particularly as Vida's contract doesn't expire until 2022.