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.
"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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