The Swans reportedly activated the £1million release clause on the 22-year old full-back which would have seen him join the Magpies but Swansea chairman Huw Jenkins insists that the young Wales international is going nowhere.
"We recognise Neil as a young, talented Welsh player that has an important role to play at this club," Jenkins said on the club's official website.
"We brought him here from Wrexham and have helped nurture his development. He adjusted very well to the Championship last season and we look forward to seeing him play a big role as we enter the Premier League.
"That is why we have rejected Newcastle's bid. I spoke to Neil's agent and he understands the situation.
"We have a big challenge ahead in the Premier League, but together we can maintain the progress made by this club over the last few years."