Eric Daniels is the current incumbent in the position and Lloyds has insisted it is behind him, but headhunting firm JCA has been appointed to draw up a potential list of replacements, reports the Guardian.
Sir Win is now said to have taken matters into his own hands through the meeting with Lord Davies, according to City of London insiders.
One told the newspaper: "It is thought that Bischoff did not offer any role to Davies â¦ but was merely trying to gauge the Welshman's interest in being considered for the role if it became free."
Earlier this year, Lloyds posted a Â£6.3 billion ($9.6 billion) loss for its operations in 2009.
The bank said in March it expects to record a return to profit this year.
By Tony Aynsley