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Diego Simeone's side are having a difficult season, eliminated from the Champions League at the group stage and battling to finish in the LaLiga top four, although they qualified for the Copa del Rey quarterfinals on Wednesday with a 2-0 win at Levante.
Speaking at a news conference on Friday, Simeone said: "A player that has a lot of enthusiasm has arrived. He is eager to rediscover his best form.
"We are working so that we can find the player he was at Lyon, where he showed his physical strength and his abilities. We will work with him to help him reach his best form and that he, in return, can give us what we want from him.
"He can play as a forward or as a second striker. I see him similar to [former Atletico forward Diego] Costa, who started out as a striker but ended up forward because of his explosive play, his strength and his good finish.. Memphis has proved, especially in France, and in his first year at Barca that he has good numbers."
Memphis, 28, had missed training with Barcelona on Wednesday as the move neared completion.
Sources told ESPN the initial fee would be €3m, with just over €1m potentially due in variables.
Depay joined Barca on a free transfer in July 2021 signing a two-year deal, having previously played for PSV Eindhoven, Manchester United and Lyon.