The Magpies' initial bid of around £6.5m was rejected earlier this month, with Twente understood to be valuing de Jong, a Dutch international who was part of Bert van Marwijk's squad at the European Championships, at closer to 15m Euros.
In the meantime, Chroniclelive reports that Newcastle are also prepared to re-enter the race to sign FC Twente's Brazillian star Douglas.
The centre-back has found himself in hot water with the Dutch outfit after failing to report for pre-season training – and has subsequently been fined by McClaren.
Finally, BBC Sport reports that Premier League newcomers Reading have signed midfielder Danny Guthrie following the expiry of his contract at Newcastle United.
Guthrie, 25, has signed a three-year contract with the Royals, and joins Russian striker Pavel Pogrebnyak among Brian McDermott's summer signings.