Suarez has been vocal in his desire to leave Anfield and move to Real Madrid this summer, citing unfair treatment by the British media as one of his main reasons.
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But Poyet, who is currently unemployed following his dismissal from Brighton, believes moving elsewhere in England's top flight should not be an option for Suarez.
"If I was Suarez, I would either stay at Liverpool or leave England for good," Poyet told Uruguayan radio station Sport890.
"Him going to Arsenal won't change a thing. If he wants to move, it's probably because of the way the media treats him as a Liverpool player, but that won't change if he moves to another club in England.
"If it's just for the sake of playing in the Champions League [possible Arsenal move], well, that's up to Luis to decide. But I'd be a lot more drastic: I'd stay put at Liverpool or leave England all together.
"I believe he has to be calm when making this decision. He shouldn't let emotions play a part. It has to be a decision that is well-thought and that will bring him good things for the future."
"Luis' numbers last season were out of this world. Liverpool, with all due respect, were a mediocre team last season, but Luis showed a great level."