The left-back, 27, said in the Sun on Thursday that he felt he and his team-mates are underpaid and that Spurs need to sign well-known players and “not ones you had to Google”.
Rose has now said the timing and manner of his comments was “ill-judged”.
“I forgive him and now we move on,” said Pochettino.
“Of course [he has a future at Tottenham]. Nothing has changed in my vision or perception here.”
Tottenham, who finished second behind Chelsea in the Premier League last season, have not signed any players so far this summer and sold right-back Kyle Walker to Manchester City in July for £45m.
Rose said in his newspaper interview that Tottenham needed to sign “two or three players”.
“My words were were not meant to offend,” added Rose on Friday in a statement released through his management company.
“For that I would like to apologise to the chairman, manager, my team-mates and the fans.”
Rose, who signed a new five-year contract last year, joined Tottenham from Leeds United in July 2007.
He has made 144 appearances for the club after becoming a first-team regular in 2013.