Earlier this week, Suarez accused Liverpool of breaking promises by preventing him from joining another club, but manager Brendan Rodgers has refuted that and forced the Uruguayan to train on his own.
Arsenal have had two bids for Suarez rejected, but Carragher does not think the striker should be thinking about turning his back on Liverpool.
“It didn’t surprise me that Brendan Rodgers has made him train on his own,” wrote Carragher in Saturday’s edition of the Daily Mail newspaper.
“I never thought Luis would down tools in the way Fernando Torres did when he wanted to leave Liverpool — his attitude over the final 18 months became progressively worse — but now Suarez has given me reason to think again.
“To see it come to this is very dispiriting. A lot of good people have stood by Luis since he joined the club. Rodgers and Kenny Dalglish could not have done more for him.
“In the darkest times, we wore T-shirts to show we were standing alongside him, even if it was ill-advised.
“So it is laughable to hear he is unhappy with Liverpool for not letting him join Arsenal.”
Liverpool’s players donned T-shirts in support of Suarez after he was accused of racially abusing Manchester United’s Patrice Evra in 2011, an offence for which he received an eight-match ban.
Carragher retired at the end of last season after making 737 appearances for Liverpool, but he remains closely involved with the club and claims he has noticed a change in Suarez’s behaviour.
“I took part in a training session at Melwood before Steven Gerrard’s testimonial last Saturday and I was alarmed by the Luis Suarez with whom I was sharing a pitch,” Carragher said.
“From his first day at Liverpool, I had been used to seeing a warrior who loved his football and charged around with great enthusiasm.
“On this occasion, however, his attitude was completely different. He didn’t want the ball and had no interest in getting involved. He stood on the periphery, with shoulders slumped.”
Speaking on Friday ahead of a friendly against Celtic in Dublin, Rodgers expressed confidence that Suarez will be able to resurrect his Anfield career.
“It’s been a difficult period for him but it’s my job to protect the group. Once he’s back with the spirit he’ll rejoin the group,” Rodgers said.
“Luis has fought for his life for Liverpool. There will come a point where he’ll recognise the club is not going to sell and then he goes on to the pitch and he’ll give 100 per cent.”