QPR are searching for a replacement for Chris Ramsey following his sacking earlier this month, with Neil Warnock installed as interim manager at Loftus Road.
Fulham are also beginning their hunt for a new head coach after Kit Symons was axed last weekend following the 5-2 defeat to Birmingham City.
A number of names have been linked with the vacant positions, but McLeish is thought to be in the running for both jobs as he eyes a return to management.
McLeish, who is currently out of work after leaving Belgian side Genk at the end of last season, has experience of winning promotion from the Championship after guiding Birmingham City to the Premier League in 2009.
Birmingham also won the Carling Cup under McLeish's stewardship, and the Scot is thought to have admirers among the QPR and Fulham hierarchy.
The 56-year-old has a wealth of experience having managed Motherwell, Hibernian, Rangers and Scotland prior to moving to England to take over at Birmingham.
After spells with Aston Villa and Nottingham Forest, McLeish took up the role at Genk in August 2014 and turned the side from relegation battlers into top-six contenders, boasting a 53% win rate, but narrowly missed out on the Jupiler Pro League Championship play-offs.
McLeish decided against renewing his contract at Genk last May and he is understood to be keen to get back into management in the UK.