Kane watched on as Lucas Moura's second-half hat-trick lifted Spurs to a remarkable 3-2 win in the semi-final second leg on Wednesday.
Tottenham were facing an exit after goals from Matthijs de Ligt and Hakim Ziyech put Ajax 2-0 up in Amsterdam and 3-0 on aggregate, only for Lucas' heroics to send Spurs into a first Champions League final on away goals.
Kane has already been sidelined for a month with an ankle injury, but the forward is hopeful he will be available for the June 1 decider in Madrid.
"Hopefully, rehab is going well," he told BT Sport when asked if he would be able to play.
"I started straight-line [running] last week. That's why I was okay to straight-line run today [in the celebrations]. I didn't want to cut corners.
"There's still a few weeks to go."
Kane lauded the character shown by Mauricio Pochettino's side, who secured their win through Lucas' 96th-minute strike.
"It's incredible. It was a disappointing first half, we let them play and they score two goals," he said.
"At half-time we just said it was 45 minutes, give everything. We didn't have to worry about being tired.
"When Lucas scored the two you think we're going to score but when Jan [Vertonghen] hit the crossbar you think that's it. Then Lucas scored with a great finish at the end. I'm speechless.
"I'm a fan first and foremost, I want to win. And as a player as well I'm watching the game thinking what would I be doing if I was out there.
"The lads dug deep, showed character, heart and passion and that's all we can ask for."