Championship - Magpies move for Taylor
The 22-year-old Wales international made 30 league appearances last season - including 26 starts - as the Welsh club clinched promotion to the Premier League via the Championship play-offs.
"We have received a proposal from Newcastle regarding Neil," Swans chairman Huw Jenkins said. "But the proposal has not been accepted or rejected at this stage.
"We have received a number of enquiries from different clubs for Neil's services and, like all bids we receive for any of our players, we will be talking it over with him when he returns from a break next week.
"Neil is a player we brought to the club this season from the Conference with Wrexham with a view to him progressing and developing with us over a number of years.
"The last thing on our mind was to sell anyone over the summer."