The Rugby Park side have targeted the former Motherwell, Preston, Derby and Nottingham Forest boss to replace Gary Locke, who resigned after Saturday's home defeat by Hamilton.
Davies has been out of management since being sacked by Forest in March 2014.
The 51-year-old's first game in charge could be the Scottish Cup tie with Rangers at Ibrox on Saturday.
Kilmarnock, who are second bottom of the Scottish Premiership, have put Lee McCulloch in temporary charge.
The former Rangers defender, who joined the Rugby Park club as a player and first-team coach in the summer, will be helped by assistant manager Peter Leven and youth coach Alan Robertson.
Kilmarnock have won only two of their last 13 Premiership matches - losing 5-1 to bottom side Dundee United and 1-0 to Hamilton in consecutive weeks - and have the threat of relegation hanging over them for the third successive season.
They have escaped the previous two years but would face a play-off against the runners-up from the Championship if their situation remains the same.
The Kilmarnock board believe an experienced manager like Davies would enhance their chances of avoiding that possibility.