Taylor told the Hull Daily Mail: "Stuart has really impressed me since I came to the club and I think he's an excellent prospect. We are talking to Newcastle and we're trying to sort something out."
He added that he is "hopeful" the 22-year-old will sign.
Full-back Carl Regan could also be set for a longer stay at the club. He's been on loan from Barnsley for the last two months and Taylor is keen to keep him in his squad.
He said: "We're close to an agreement and again it would be nice to think something will be sorted out this week."
The Tykes are in administration and are happy to get Regan off their books. City have offered him a contract until the end of next season.
Meanwhile, Paul Musselwhite will put pen to paper on a new contract. His current deal runs out at the end of the season, but the goalkeeper will sign a one-year extension.