The in-form midfielder was withdrawn at half-time at Wembley and replaced by Mousa Dembele, having gone down holding his ankle before the break.
Winks' absence could limit Gareth Southgate's options even further after Dele Alli was forced off due to injury during Tottenham's victory.
"He twisted his ankle. It's a little bit painful," Pochettino said.
"We must assess him with our medical staff and with the national team medical staff too. He's so excited to play for the national team. I don't know (if he can play). It's about him and both (teams of) medical staff.
"Dele Alli against Manchester United finished the game feeling his posterior tendon, close to the hamstring, and then played 90 minutes against Real Madrid.
"It's a very, very, very small thing. It was a massive risk (to play him) again. We're sure he'll be ready for Arsenal (after the international break). We hope he's going well in the next few weeks and will be available.
"We wanted after 75 minutes to protect (the substituted Harry) Kane. He's okay, he's going with the national team.
"The Christmas period is so tough for the players. You cannot do too much; only have good communication and a good relationship with the national team and explain the situation with everyone and trust they will manage them in a good
way.
"We knew it would be a massive challenge for us to be ready again and fresh after (Wednesday's 3-1 defeat of Real Madrid). But I am very pleased. The result was fantastic even if it wasn't our best performance."