Kane struck his third hat-trick in his last nine appearances in last Sunday's 4-0 win over Stoke City but appeared to suffer a minor knock in the closing stages of the match, with Heung-min Son coming onto replace him.
The England international has been training this week but Pochettino says the club will make a late decision on him ahead of Sunday's trip to Goodison Park.
"The first moment he said it was good. When after a few minutes he tried, I asked. In that moment we made two changes. He said I am okay," Pochettino told a press conference, Sky Sports report, when discussing Kane's knock.
"But at the end of the game when it got cold, it appeared more painful. But I trust that he can play Sunday. If not, if he is a risk, he will not be in the squad."
Both Alderweireld and Vertonghen were also substituted during the Stoke win last Sunday with respective groin and ankle knocks. Mauricio Pochettino described their injuries as "nothing important" when speaking after the comprehensive victory over the Potters but the club are yet to make on a decision on whether the Belgian duo will be ready for Sunday's trip to Goodison Park.
Rose meanwhile has missed his side's last six appearances in all competitions after picking up a knee injury in the goalless draw against Sunderland at the end of January. The England international had been perhaps Tottenham's best player of the season before that setback, with his presence down the left greatly felt during his absence.
Ben Davies has deputised in place of Rose and will continue to do so with their first-choice option still "a number of weeks" away from a return. Lamela meanwhile remains a long-term absentee having not played for the club since mid-October due to a hip problem, but recently returned to light training on the field. A time-frame for his return remains a number of weeks away, however.