The 31-year-old Belgium worldwide was forced off in the first half of Tottenham's 3-2 win at Wolverhampton Wanderers and missed Tuesday's 2-1 Champions League victory over Dutch side PSV Eindhoven.
Spurs have been in the national stadium since the beginning of last season and recently committed to staying there until at least January following significant delays to their new ground.
"After it's about the evolution, it's very dynamic the evolution and how he's going to recover".
Unsurprisingly, Pochettino said he would prefer to keep his players at the club, but added how he managed them would be crucial.
"It is always about the patience, how he is going to react to the injury but we hope he is going to be ready by the early January".
"We need to be grateful for Wembley because without Wembley now we play where?" he said.
The Serie A champions have often been left with only three fit central midfielders as a result, and Corriere di Torino reports that they want reinforcements for the second half of the campaign.
However, with a challenging trip to London rivals Crystal Palace to come on Saturday, Pochettino and his medical staff will hope there are no setbacks with Dier and Wanyama, who are both slated to return to the squad.
"It's a massive effort for the players".
Pochettino said: "I saw him this morning, I asked for his perfume because he has very good perfume".
"We will see what the evolution is, it may be a situation that can change quickly and be positive". It will be tough to play two games. It is play, play, then move them there and play again.