Cardiff's hopes of holding on to Ledley appear to have dropped after missing out on a place in the Premier League.
The Bluebirds were beaten 3-2 by Blackpool in the Championship play-off final at Wembley last month.
Ledley was on the score sheet to put the Welsh side 2-1 ahead, but his day ended in tears at the final whistle.
Cardiff City's new board said they would offer the player an improved offer to stay.
His negotiations with his hometown club had already dragged on for 18 months, but he said he would only make a decision on his future once he knew which division the Bluebirds would be playing in next season.
If Ledley decides to move to England's top flight the buying club would have to pay a compensation fee to Cardiff because he is under 24.
Cross-border transfers to Scotland, Northern Ireland or anywhere abroad do not require a fee once a player turns 23.
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