Kane, who has not played since April 9 due to an ankle injury, was at the Johan Cruyff Arena as Lucas Moura scored a last-gasp winner to complete an incredible comeback, booking Spurs’ place in the final on away goals after a 3-3 draw on aggregate.
The England striker ran onto the pitch at the full-time whistle to celebrate with his elated team-mates and manager.
The final takes place in Madrid on June 1 and Kane is hoping that he will play a part.
“Hopefully [I will be fit in time],” Kane told BT Sport. “Rehab is going well so far. I start straight line running this week.
“So far so good, we will see. If it carries on like it should then I hope to be, but there’s still a few weeks to go.”
Spurs looked to heading out of the Champions League as they trailed Ajax 3-0 on aggregate at half-time of the second leg in Amsterdam, but a second-half hat-trick from Lucas saw Spurs reach the final in thrilling style.
“Just incredible,” said Kane. “Disappointing first half, we didn’t start well and let them play and they scored two goals. At half-time we said it’s 45 minutes to give everything. It could be our second to last game of the season, we don’t have to worry about being tired, just give everything.
“When Lucas got two then you’re thinking we are going to score, then Jan [Vertonghen] hit the crossbar and you think that’s it, it’s not our day, but Lucas scores a great finish at the end. Speechless really.