The Belgian centre-back has not featured for the Lilywhites since limping off against Real Madrid in his side's famous 3-1 win on November 1.
Alderweireld was ruled out of action until February due to the severity of the damaged hamstring, but Pochettino hopes to have one of his key players back slightly ahead of schedule in time for the fourth-round tie on January 27.
Asked if Newport away was the target, Pochettino told the assembled press: "Maybe, maybe. We are going to have all week for training so we can see if he can arrive in a good condition to play. We are going to assess day-by-day next week.
"Today all is okay with him, maybe tomorrow he can integrate himself in some exercises with the group and a little bit step-by-step to build his fitness to be available again.
"He will do a warm-up with the team, the first exercise and to be integrated gradually with the group and we will see next week if he can improve and then be ready as soon as possible."
Alderweireld is out of contract at Spurs at the end of next season, but talks over a new deal are said to have hit an impasse.