McLeish revealed the the Chilean international suffered the setback during the home clash with Wigan nine days ago, ruling him out of the weekend defeat against Everton.
The Blues boss admits the Blues medical staff have advised him to tread carefully with the 26-year-old, who signed in the summer from Club America. But he is keeping his fingers crossed Beausejour, who has yet to start a Barclays Premier League game, will be available for the trip to Arsenal on October 16th.
McLeish told Press Association Sport: "Beausejour had an injury which he further aggravated last week so he was ruled out completely for the weekend. He got a nick in his medial ligament against Wigan when he came on.
"The doctor ruled him out immediately and said 'if you want to get anything out of this guy this season, you've got to be careful.'
"I've got to respect the medical staff. But hopefully the fact it is the international break will give him the chance to make a full recovery."
Source: PA
Source: PA