Wolves’ precarious financial position means that Kilman’s future at the club has been under the scanner in the ongoing transfer window.
Napoli have already failed with a €35m bid for the centre-back and are pondering over tabling a new offer.
However, Wolves and Kilman are not desperate to part ways at this stage and the club offered him a new contract on Thursday.
And it has been claimed that there is increased optimism that the defender will sign the new deal on offer.
Kilman is not pushing for a transfer away from Wolves at the moment despite the real interest from Napoli.
Wolves have been prepared to sell him if their valuation is met but are also happy to keep him.
The club are now hopeful that Kilman will eventually sign a new contract and end speculation over his future this summer.
Wolves signed Kilman from non-league side Maidenhead United in 2018 and he has progressed to make more than 100 appearances in the first team.
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