Taylor, 25, joined County in May from Macclesfield, but has yet to start in the league.
The move has left Newport with only one goalkeeper in Joe Day, but manager John Sheridan is confident he will quickly fill the vacancy.
"Rhys is too good a goalkeeper to be kept on the bench and not playing football," said Sheridan.
"Financially it might enable me to bring someone else in, to try and fill in a different area."
On Saturday Newport host fourth-placed Northampton Town hoping to win their third game in a row for the first time since last December, 2014.
The Exiles are also seeking their first home win since March.
Taylor's move means County are currently without a second goalkeeper.
Sheridan added: "Rhys' move to Wrexham happened quickly... but we'll definitely have a keeper by the middle of next week.
"It's only one game. I'm not really worried."
Sheridan also confirmed that defender Janoi Donacien has extended his loan spell from Aston Villa until January.
Donacien has featured in all four games since the Irishman replaced Terry Butcher as manager at the beginning of October.
Meanwhile Newport have no injury concerns ahead of Saturday's game.
Newport County goalkeeper Rhys Taylor has joined Wrexham on loan until January.