The report in The Mirror comes in the wake of another recent rumour afoot that Brum boss Alex McLeish was set to return with an improved offer for the player after Fulham rejected a £1.5 million bid.
29-year-old Konchesky was born in Barking and played for the Hammers from 2005-2007 and it is thought he would prefer a move back to familiar surroundings.
With just one-year remaining on his Fulham contract the player is said to favour a fresh challenge and has told the club he does not want to re-sign.
Joining West Ham in July 2005 for £1.5 million Konchesky made an impact over 45 Premier League and cup appearances, seeing West Ham to a ninth-place finish and the 2006 FA Cup Final.
However, he lost his place in the first team during 2006-07 as West Ham narrowly avoided relegation to the Championship and eventually left in July 2007.
After signing a four-year deal with Fulham following a £3.25 million transfer he rubbed salt into West Ham's wounds by scoring from 35-yards in a 3-1 defeat in January 2009.
And while The Mirror insists that Birmingham City are still favourites to secure his services his personal preference is to move from South to East London.
Another intriguing aspect of the transfer saga is remaining cross-boardroom tensions between Upton Park and St.Andrews given David Sullivan and David Gold's former tenure at Birmigham.