The Guardian says the development became known after Leeds United officials sought information on Strachan's situation at St Mary's.
Leeds chief executive Trevor Birch is now keen to know whether the ex-Leeds idol is more attainable than could have been first anticipated.
However, Leeds insiders are not that confident of convincing Strachan to leave the safe haven of St Mary's.
Strachan is yet to finalise his Saints contract beyond this season, but a source close to Elland Road told the Daily Echo: "I'm sure they would love to be able to get Gordon Strachan, he was here for a long time and remains extremely popular.
"But I can't imagine for a moment that he would want to come here with the state the club is currently in."