Taylor had a £250,000 offer for Swindon's Sam Parkin turned down and so he is now concentrating solely on 27-year-old Peacock instead.
He said: "Lee Peacock is one of the other players on my list and I know he'd do a really good job for us.
"He's proved himself time and time again at Bristol and they would probably have offered him a really good deal - if they'd won promotion.
"However, it looks like he's available on a free and that could make things easier for us - if we decide to take things any further.
"We'll be talking to his agent to see exactly what the situation is but he's a player I already know all about."
Meanwhile, City will not be following up interest in Yeovil's Gavin Williams or Torquay's David Graham.