Liverpool are understood to have been joined by Atletico Madrid as well as Sevilla in the race to sign Vida, who is valued at £30m by his club side.
According to reports in Turkey, the Reds have doubled down in their effort to get the signature of one of the World Cup's star player.
With demand at a high, though, having also been linked with Leicester City plus Everton, Liverpool are expected to have their most recent bid cast off.
A Besiktas spokesperson is quoted by the publication as saying: “We have received offers for Vida and we are considering them but that does not mean he will definitely leave.
“Every player has a price, we will make a final decision in accordance with our budget and interests.”
The 29-year-old played in six of Croatia's seven World Cup games, counting the full 90 minutes of Sunday's 4-2 loss to France in the final.
This article was most recently revised and updated 3 years ago