McLeish was confirmed as the new man at the helm at Villa Park on Friday and has already begun making plans for his moves in the transfer market.
Manchester United stalwart Brown has been tipped to leave Old Trafford this summer after 15 years with the club.
The versatile defender made just 15 appearances last season and has already been linked with a switch to Sunderland under former United star Steve Bruce.
McLeish believes that the 31-year-old former England international could be a useful addition to his Villa squad and is considering an approach.
Newcastle full-back Danny Simpson has also been linked with a possible switch from St James' Park to Villa Park in recent times.