Kane netted four in a 10-0 win over San Marino in a World Cup Qualifier.
He now has 48 England goals, level with Gary Linekar and only five behind Rooney's tally of 53.
Kane has 67 England caps, and Southgate thinks that if he gets close to Rooney's 120, he will smash the former Manchester United star's record.
Asked about the matter by reporters, Southgate said: "I think that would definitely be one of his goals [setting a much higher goal record].
"The remarkable thing is we haven't given him the opportunities with nights like [San Marino] too often.
"Certainly the other three games of this magnitude in this group, we didn't give him the minutes. He understood that, of course. He has got huge personal ambitions as well as the team ones. He showed again the clinical nature and mentality he has got.
"There is no reason he can't. It's not is he going to break the record, it is when, and what might he do once he is beyond that point."