Meulensteen has confirmed that the London club have lodged a bid for the Wales international, who has failed to regain his place in the Swansea side since recovering from a serious knee injury.
Ben Davies has established himself at left-back in Michael Laudrup's side and Taylor has become frustrated with a lack of first-team football.
That has alerted Fulham, who have reportedly offered £4 million for the 24-year-old former Wrexham man.
Meulensteen revealed that Taylor caught his eye when he was on the coaching staff at Manchester United and the Dutchman is determined to get his man.
"We have put a bid in. It's on the table. It's simmering," Meulensteen said.
"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."
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