Eriksen suffered cardiac arrest in Denmark's clash with Finland, causing him to be resuscitated on the pitch.
The 29-year-old is alive and well and recovering in hospital.
"It's been difficult," England captain Kane said. "Obviously to watch it last week and see it happen was really hard.
"Thankfully Christian is alive and seems to be doing well. I've not had contact with him personally but my wife has spoken to his wife and from what I understand, everything's going in a good way.
"Obviously we've had to try and switch off from it and focus on the game, but of course Christian is in all of our hearts; not just our team, but the whole football family.
"It's a reminder of how quickly things can change in life; never to take things for granted. We want to wish him well, wish him all the best.
"I hope to see him soon personally. Hopefully after this tournament we get to do that."