The 29-year-old, who has one year to run on his current deal at Craven Cottage, has been a virtual ever-present in the Fulham team since joining from West Ham in 2007.
But it is believed Roy Hodgson is keen on signing Aston Villa left-back Nicky Shorey on a permanent basis after joining the club on loan in January, suggesting the Cottagers boss would be willing to do business for Konchesky.
That has alerted the likes of Birmingham and Bolton and the West Midlands club have stepped up their interest in the twice-capped England international.
McLeish's side were the surprise packages in the Premier League this season, finishing the campaign in an impressive ninth position and have been in the hunt for a left-back after playing the majority of the season with centre-half Liam Ridgewell filling in.
David Murphy, the usual left-back, has been injured all season while second-choice Gregory Vignal has not managed to secure a new deal despite spending a season-long spell at City this year.
Blues have made rapid moves in the transfer market as they prepare for their second successive season in the top-flight with Manchester United goalkeeper Ben Foster due to sign at St Andrew's in the coming days while former Barcelona youngster Enric Valles will also been arriving in July.
McLeish has reportedly offered £8million for Wigan winger Charles N'Zogbia while the Scottish boss has also been linked with moves for Carlton Cole and Kris Boyd in recent weeks.