The Blues were thrashed 4-1 by Watford at Vicarage Road on 5 February – the club's second successive defeat by a three-goal margin after the 3-0 defeat against Bournemouth.
Conte, who had asked for a public vote of confidence from Chelsea's owners before the Watford game, said he did not fear losing his job and accused his team of playing with "no personality" at Vicarage Road.
Carragher said he could not see Conte lasting too long as Blues boss considering how often the club have changed managers in the past.
"There's a good chance he could go before [the West Brom game]," the former England international told Sky Sports.
"Not just because of how Chelsea are playing of late but because of Chelsea's history of changing managers when the club fear they are not going to make the Champions League.
"They did it with [Andre] Villas-Boas, [Roberto] Di Matteo, [Jose] Mourinho as well."
Conte won the Premier League title last season with Chelsea, but the Blues are currently 19 points behind league leaders Manchester City. The Italian insisted last week that he intended to see out the remaining 18 months on his contract at Stamford Bridge.
Conte's relationship with Chelsea owner Roman Abramovich has come under scrutiny in recent weeks, amid reports the Russian oligarch was unhappy at the manager's outspoken comments regarding the club's transfer strategy.
"It looks to me when I hear him speak in press conferences I wouldn't say he wants to be sacked but it doesn't look like it'd be the worst thing in the world if it happened," Carragher added. "It's almost like he's testing the board at Chelsea.
"He's prodding and poking them in press conferences and interviews and that will not go down well with Roman Abramovich."