City it's reported are said to be considering an improved £400,000 bid.
Huddersfield born Jerome joined City from the North East, he moved from Middlesbrough in July 2004.
City are also thought to be considering the futures of high earners Tony Warner, Alan lee and Andy Campbell. City could look to assist with their wages if a new club takes them on.
The speculation thgat Richard Langley could still return to QPR is still there and it seems certain Cardiff will lose Danny Gabbidon. James Collins and Peter Thorne are other high profile players who could move.
Of the out of contract players it looks like Neal Ardley, Darren Williams, Toni Koskela and Willie Boland will be offered new reduced deals while there's little hope for Michael Boulding, Gary Croft and keeper Arran Lee Barret as all three are set to be released. Tony Vidmar has moved to NAC Breda in Holland.