Highly rated Davies is valued at around £5m, and Everton's new bid comes after Barry Ferguson rejected the possibility of moving to Goodison.
Keith Wyness has said: "We have been in discussions with Tottenham over the weekend but no conclusion has been reached."
Davies can play on the right of central midfield.
More news as we get it, but the latest word is that Everton's latest offer has been rejected. Expect more news this afternoon.