The 24-year-old left-back has found opportunities limited after a broken ankle suffered last season allowed Ben Davies to take his place.
Fulham have already had a bid of ?4million rejected for the Wales international, but are believed to have made a new offer.
Laudrup made it clear Swansea will not sell Taylor unless he can bring in a replacement, and Press Association Sport understands the Welsh club have had a bid of ?2.5million rejected for Leicester's Jeffrey Schlupp.
"Neil is here now and I don't want to go into discussion if he's staying or leaving because you can never know that," Laudrup said.
"What I've said about Neil is that I fully understand his position, especially after going through what he's been through.
"But we will only sell him if we can get a replacement.
"Until midnight, everything is possible
That's why I don't want to go into all the rumours."
A potential complicating factor in any move is that Taylor is currently with his wife, who is in labour.
But Laudrup would not confirm whether a second bid has been made by the Cottagers.
"I don't want to comment on that," he said
"When I talk about facts it is when something is done one way or another
We cannot inform you every time."
Swansea are also known to be interested in Norwich duo Anthony Pilkington and Wes Hoolahan, while a deal to bring Falkirk youngster Jay Fulton to south Wales is set to be completed.
One player leaving Swansea is striker Rory Donnelly, who has joined Coventry on loan until the end of the season.
Source : PA
Source: PA