The South Korean was substituted by the Italian towards the end of Tottenham's 2-0 defeat to Chelsea in the first leg of their League Cup semi-final on Wednesday.
Conte said he was not aware of the injury at the time, but that Son was now likely to be missing until after the international break, which starts later this month.
That would rule him out of the second leg of the League Cup semi-final and Premier League matches against Arsenal, Leicester and Chelsea as Spurs chase a top-four spot.
Son, who is the club's top scorer in the Premier League this season, could also miss South Korea's World Cup qualifiers against Lebanon and Syria.
"It was a strange situation because against Chelsea, I made two substitutions together, with Lucas Moura and then Son. But not for injury, the substitutions, but only to try to give them a bit of rest," Conte said on Friday.
"Then the day after, he felt a bit of pain in his muscle, in his leg. He had a scan and now we have to make the right evaluation from the doctors.
"He has an injury. I don't know, but probably he will stay without training sessions until the end of this cycle, for the international break."
Conte also hinted that Harry Kane would be rested for Sunday's FA Cup third-round tie against Morecambe.
Conte pinpointed the gulf in quality between his squad and Chelsea's for the disappointing result in midweek, saying he would hold talks with chairman Daniel Levy on the possibility of new signings in the January transfer window.
"I'm sure that we want the best for Tottenham," said Conte. "The first person who wants to improve the situation is for sure our owner.
"I will have time to speak, also with the people who work within the club to try to understand well the situation, to understand if there is the opportunity to improve this squad in this transfer market."
Conte also said the door was "always open" for former Spurs player Christian Eriksen as the Denmark international targets an unlikely comeback following a cardiac arrest during a Euro 2020 group match last year.
"To see him again ready to play football is great news," said Conte, who signed Eriksen when he was Inter Milan boss in 2020. "I think for Christian, the door is always open."