Pochettino has earned widespread acclaim for taking Spurs to their first Champions League Final and securing a fourth consecutive top-four finish this season without signing a single player.
He has done so while leading the migration from the old White Hart Lane to the new Tottenham Hotspur Stadium, via a longer-than-expected stay at Wembley.
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However, the Argentine has hinted that he could leave the club if they win the Champions League Final, and Rose believes the time is now for the Spurs board to back their manager.
"They [the board] must listen to the manager. He said recently that he needs to know what the next chapter of the club is. I fully agree with everything he says," Rose told Talksport.
“Not many people could operate in the circumstances that he’s done. We had no stadium, we’ve not signed anybody.
“Not many managers would have been able to do what Mauricio has done, in terms of securing Champions League football four years in a row and finishing second, third and fifth.
“We definitely have to listen to what he thinks and what he wants and hopefully we can give him that.”