The Israeli, who now coaches Partizan, lasted just 58 games under owner Roman Abramovich despite leading the west London outfit to runner-up spots in both the Premier League and Champions League - losing out to Manchester United both times.
Chelsea currently lie fifth in the league, 17 points adrift of leaders Manchester City, and have endured a run of poor results, including last weekend’s draw at Stamford Bridge to Birmingham City in the FA Cup.
Their campaign so far looks set to be the worst of their Russian owner's tenure and an away trip to Italian side Napoli looks unlikely to ease the pressure on the shoulders of the former Porto coach.
Grant is one of four managers at the club that have been axed by Abramovich and the 57-year-old has warned Villas-Boas that the same fate could await him at any moment.
"My advice to Andre [Villas-Boas] is to try and do his job and win games. What will happen will happen,” Grant told reporters.
“In the old times, Roman [Abramovich] didn’t have much patience but now I think he’s learned something and is a little bit [more patient].
“I don’t know for how long though. It’s important for any manager to realise that any day could be your last.”