Leonard is available on a free after seeing out his contract at Monaco.
Blackburn have held a long-term interest in the player after failing to secure him last January.
The Ewood Park outfit have rekindled their interest in the full back after talks stalled over the signing of Slovakian international Vratislav Gresko.
Rangers are ready to hijack Rovers' attempts to sign Leonard as they are in the market for a new left-sided full back.
Having lost Jerome Bonnissel and Arthur Numan in the summer Ibrox boss Alex McLeish is short of options at the back.
McLeish is scouring the transfer market for a left back and with Leonard being available on a free transfer he is an attractive option.
Atletico Madrid are looking for defensive cover with veteran defender Sergi coming to the end of his career and they are considering a move for Leonard.
Meanwhile, Rovers have refused to rule out a move for Parma defender Gresko.
The deal hit the buffers last week with the defender's demands reportedly being too excessive, but Rovers chief executive John Williams admitted they are still in talks with the player.
"Although the deal did hit the rocks, as it were, it's certainly not dead," Williams told the club's official website.
"We are still in negotiations with Gresko and with Parma. For the moment, let's just say, we are still talking."
