England international Rose, out since January with a knee injury that required surgery, came on as a late substitute in the Champions League clash at the Bernabeu on Tuesday.
With Ben Davies having missed Tottenham's last two games through illness, Rose's return to fitness is timely but Pochettino will not rush the 27-year-old back.
"Danny is still not ready to start, he needs to build up his fitness," Pochettino told reporters on Friday. "But we are so happy because he is doing very well and for him to be on the bench on Tuesday was good for the team."
Rose could start against West Ham United in next week's League Cup clash, however, according to Pochettino.
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"Now the process is to build up and get better every day. We'll see on Wednesday whether he has the possibility to be in the starting eleven but that depends."
Rose was given a warm reception by the travelling Spurs fans on Tuesday on his first appearance since criticising the north London club's transfer policy and wage structure in August, prompting suggestions he had played his last game for the club.
"He is happy, his behaviour is fantastic. You could see he helped the team in the final few minutes against Real Madrid," Pochettino said. "His mind is clear that what happened is in the past and we need to move on."
Midfielder Erik Lamela, another long-term absentee, could make the bench against West Ham, Pochettino said.
First-choice central midfield partnership Mousa Dembele (hip) and Victor Wanyama (knee) are both ruled out and are causing Pochettino concern, with no likely return date set.
"We are concerned about Wanyama's situation. We must wait for when he sees the specialist and then we'll see what happens," Pochettino said.
Dembele injured his hip in training before the Real Madrid match and is unable train at the proper intensity, according to Pochettino's right-hand man Jesus Perez.
Third-placed Tottenham face a testing period with Liverpool at Wembley on Sunday and Manchester United away the following week with West Ham squeezed in between. They then host Real Madrid in the Champions League return.
"It's a period for us to learn a lot," Pochettino said.
(Reporting by Martyn Herman Editing by Jeremy Gaunt)
