Palmer, 21, signed a seven-year contract with the option for a further 12 months on Deadline Day as the final player to form part of Chelsea’s radical squad overhaul over the last three windows.
The expectations are high in west London after Chelsea broke the world record spent in a single summer and for three consecutive windows.
But Pochettino is urging caution with Palmer as he adapts to life at his new club.
The Blues boss said: “We need to be careful with the young guys because they need time to settle. He is from Manchester and is going to move to London.
“London is different, the club is different, the culture is different. He needs to first of all, before he starts to perform, settle and feel his space.
“We are not going to put pressure on him to perform. He needs to be happy and calm and find space in this team.”
Palmer started just three Premier League games for City and struggled to establish himself at the Etihad Stadium.
He was, however, part of a successful winning machine that secured a Premier League, Champions League and FA Cup treble last season. Palmer also won the Under-21 European Championship with England over the summer.
Pochettino is excited to sign the winger, who can also play as a central attacking midfielder, adding: “He fits the project and he is a young and talented player.
“Of course, he also decided to come because he expected to play more and be important here. That’s not the most important thing, the most important thing is that he wanted to come because he sees Chelsea as a project for him to improve his game and to maybe be more involved in every single game. But, yes, I think the quality is there. He has great potential.”
Chelsea have also signed record-breakers Moises Caicedo and Enzo Fernandez in the last few months, along with big-name players like Mykhailo Mudryk, Romeo Lavia and Christopher Nkunku.
The sheer numbers of players moving in and out of Stamford Bridge has raised eyebrows but Pochettino defended the approach and said the spending will soon slow down.
He added: “I think we need to explain the reality in the same way we sign players, we sell players. It is more about net spending than really the money you spend because that’s the balance the club is now trying to get.
“We are sporting professionals who advise them and give vision but, of course, it is up to them.
“This is a very special situation for the club from the beginning but, for sure, the club won’t keep doing for years what they have done in the past.
“We need to settle all the players who have arrived late, create spaces and build a solid structure to perform after.
“We are in a good way and we need to recover our injured players and those who have arrived late to show good form.
“We are working really hard to try to be at our best. For sure, with time, it is a matter of time but we have the quality and we are going to perform the way people expect from us.”