Vida has established himself as one of the most solid centre-backs at this summer's World Cup, helping Croatia to a shock appearance in the final of the tournament alongside current Liverpool player Dejan Lovren.
The Reds are understood to have already seen a £15.9m bid for the defender knocked back, but Fanatik reports that they are now willing to increase that offer.
Liverpool are expected to return with a bid of around £17.7m, although that figure is still thought to be some way below Besiktas's valuation of the player.
The Turkish outfit are reportedly holding out for a fee of £30m, having seen the 29-year-old make it all the way to the World Cup final.
Vida only joined Besiktas on a free transfer from Dynamo Kiev in January, but Besiktas are willing to move the defender on if they receive a suitable offer.