The England captain started and came close to scoring in Sunday's 1-0 group stage win over Croatia but failed to find the net before being substituted on 82 minutes.
Kane is regarded the first name on Southgate's teamsheet by many but comes into the tournament on the back of a long, hard season at Tottenham and is facing competition from Dominic Calvert-Lewin.
When asked whether he considers himself a certain starter, Kane told ITV Sport: "You will have to ask Gareth but I never look it at it that way. We have great competition for places.
"Dominic has come in and done great. For me it is about training how I train every day try and be the best and try and improve every day and when I play try and give my best for the team.
"I think that is what everyone is doing at the moment. I am sure there will be rotation among all the players in this tournament and everyone is waiting for the chance to try and make an impact."
Southgate may choose to heavily rotate his side for the third group game against Czech Republic if his side have already secured qualification by that point with a positive result against Scotland on Friday night.
Kane was named among the substitutes for England's final group stage game at the 2018 World Cup against Belgium, when a place in the knock-out stages was already assured.
On the prospect of another turn on the bench, Kane said: "Well, that is not my decision. I will train as hard as I can to play and that is the aim to try and lead the boys out and win the games."
Kane faced Croatia less than 24 hours after his former Tottenham team-mate Christian Eriksen suffered a cardiac arrest during Denmark’s defeat to Finland in Copenhagen.
Eriksen is now in a stable condition in hospital and posted to Instagram revealing that he is “fine, under the circumstances” on Tuesday, but Kane believes that the horrific incident put the summer’s football into perspective.
“I texted him, but haven’t got a response yet, obviously,” he said. “He is going under some tests. My wife managed to speak to his wife and she said, obviously, they are doing ok, they are going through the checks.
“It was a scary time for him and his family as well. It was dreadful to watch. Like I said at the weekend, I am wishing him all the best and his family all the best, and I am hoping he makes a speedy recovery.
“Sometimes we take things for granted and we take football for granted and life for granted. That was a reminder that things can be taken away so quickly.
“Like we said, it was a dreadful thing to watch and experience. For us it is just about enjoying this experience, enjoying this tournament and going out there with no fear.
“We want to win, we are all winners, we are passionate to win - just express yourself and enjoy the occasion because they don’t come around too often.”
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