He is currently playing with the South Yorkshire side on a season-long loan, a similar arrangement to last season when he spent much of the season with fellow third division outfit Carlisle United.
Green has started all three of Hull's games so far this season, including the defeat at Exeter City on Saturday where he scored and was then sent off.
Now Hull Chairman, Adam Pearson, says that he is keen to keep Green in Yorkshire.
"We're doing everything we can to keep Stuart. We know Newcastle have received a number of offers and our's matches the best," he is reported as saying yesterday.
"We've been assured that if they do decide to sell him, we will have the first option but it might not be as straightforward as that. There are other things to consider. For example, Stuart has family and friends in the Carlisle area.
"At the end of the day, everything depends on what Newcastle decide. It's by no means certain they will want to sell him and if that's the case we're hopeful he will continue his loan spell with us. I want to assure our fans we're doing everything we can."
Carlisle United would also be able to match any offer made by Hull. New chairman John Courtenay, who runs the UMBRO franchise in Ireland, is keen to bring in new players to the Cumbrian side.