The defender – who only joined the Tigers over the summer from Birmingham – has made a fine start to the season under the guidance of Steve Bruce.
His impressive displays have been key in the Humberside outfit’s solid start to the season, which has seen them avoid the tag of being a side expected to be relegated at the end of the Premier League campaign.
Davies’ consistent displays are believed to have caught Roberto Martinez’s attention, with the Toffees chief keen to explore the possibility of a deal.
The Merseysiders have been, arguably, the season’s surprise package, but their Spanish manager is still looking at expanding his options next month.
The 28-year-old enforcer is thought to be on his wish list, finding a long-term replacement for Sylvain Distin a pressing concern.
Hull are unlikely to be willing to discuss an arrangement, but they could be forced into letting their man leave if he pursues a switch.
Davies’ impressive form has led to some discussion of a potential England call-up for the 2014 World Cup.