The Cottagers have been linked with a move for the Wales international during this transfer window and have already had one bid turned down by the Swans as boss Michael Laudrup is reluctant to sell unless he can bring in some cover.
It is thought Taylor is keen to leave for Craven Cottage as he wants regular first-team action and he has become frustrated playing second fiddle to Ben Davies at the Liberty Stadium.
And Meulensteen, who revealed that former Manchester United boss Sir Alex Ferguson was keen to sign the then youngster from Wrexham before he suffered a serious ankle injury, has now broken his silence by confirming that an offer has been made and the Londoners are hopeful of securing the player's signature.
He said: "We have put a bid in. It's on the table. It's simmering.
"I rate him. I rated him when I was at United. The boy was very unfortunate. He had that horrific injury at a time when he was really playing at his best and he had top four qualities."