Enrique admits he was furious at being told he was surplus to requirements by new Liverpool manager Brendan Rodgers earlier this season.
But he insisted he would never react like Odemwingie, who drove to QPR in a bid to force through a deadline-day transfer.
The Spanish star, 27, said: "I'm not the type of player who would wind him up.
"But there will be songs and he will get it from other players.
"They will try to make him angry.
"I'm not this type of player. But a lot of players will do it."
Unlike Nigerian ace Odemwingie, Enrique knuckled down at his club and, along with fellow Anfield 'flops' Stewart Downing and Jordan Henderson, has forced his way into Rodgers' plans.
He said: "For Odemwingie, it's his mistake - it's not my problem. I would never do that."