Press Association Sport understands that the Magpies submitted a formal offer amounting to more than four million euros for the 24-year-old and are now waiting to hear back.
However, it is also understood that Twente have placed a higher - if not much higher - valuation on the player, who is being courted by a series of other clubs.
The Magpies discussed Douglas with Steve McClaren's club during their aborted bid to snap up striker Luuk de Jong, but were relaxed in their pursuit with the player needing to remain in Holland until late next month to ensure he is eligible to represent his adopted country after being granted residence.
But they have now stepped up their interest once again as they attempt to close in on their major targets for the summer.
McClaren does not want to lose Douglas and will be keen to recoup as much money as he can if he is to leave.
However, Newcastle owner Mike Ashley has already demonstrated his unwillingness to bow to Twente's demands in recent weeks and is unlikely to bend too far once again.
It has proved an eventful few days for the Tyneside club with confirmation of their interest in Liverpool striker Andy Carroll and their ongoing attempts to sign Lille full-back Mathieu Debuchy having propelled them into the headlines once again.
But in all three cases, they are not alone and their determination to stick to their stringent recruitment policy is likely to be tested to the full as they attempt to secure the deals.