The 27-year-old has spent the last six seasons at Wolves and has established himself as an integral first-team player in recent years, making 151 appearances for the club in all competitions.
Kilman is a fans' favourites at Molineux and was appointed Wolves captain last summer shortly before signing a new five-year contract, keeping him tied to the club until June 2028.
The defender was a mainstay in Gary O'Neil's side last season, playing the full 90 minutes in all 38 Premier League games as the Old Gold secured a 14th-placed finish.
Kilman's consistent form is understood to have caught the eye of a number of English clubs, including West Ham United, who are now managed by former Wolves boss Julen Lopetegui.
Newcastle, West Ham fail with Kilman bids
A recent report claims that West Ham have tabled their first official bid for Kilman, believed to be worth £25m.
However, The Telegraph states that the Hammers' offer was rejected by Wolves, who have since received a new proposal from fellow top-flight team Newcastle.
The report adds that the Magpies have made a player-plus-cash bid for Kilman, including home-grown midfielder Elliot Anderson, but the offer has been turned down by Wolves.
Newcastle are said to have submitted their offer for Kilman in the last 24 hours and must improve their bid if they are to stand any chance of luring him to St James' Park.
It is also understood that Magpies boss Eddie Howe would be reluctant to part ways with 21-year-old Anderson, who made 26 first-team appearances during an injury-hit 2023-24 campaign.
Both Newcastle and West Ham, who are prioritising the addition of a new centre-back this summer, will now weigh up whether to make an improved offer for Kilman, who has also attracted interest from Manchester United.
What is Wolves' stance of Kilman future?
Wolves have no interest in losing Kilman to a fellow Premier League side, especially given that the defender only signed a new long-term contract last summer.
The Old Gold were required to sell some first-team stars last summer, most notably Matheus Nunes, Ruben Neves and Nathan Collins, to avoid breaching the Premier League's Profit and Sustainability Rules.
According to The Telegraph, Wolves are now under no pressure to generate funds to ease their financial concerns, but they may need one big-money sale to allow O'Neil to bolster his squad.
Kilman was allegedly the subject of a £30m bid from Serie A giants Napoli last summer, but Wolves now value their skipper - who joined the club from non-league outfit Maidenhead United for just £40,000 in 2018 - in excess of £40m.