Rooney, 33, returned to make his 120th and final international appearance in a 3-0 friendly victory over the United States at Wembley on Thursday.
The DC United striker has scored 53 times for England - 34 more than Kane, who is eight years his junior.
"I look forward to presenting him with the trophy," said Rooney.
Kane and FA chairman Greg Clarke made a presentation to Rooney before the game.
Rooney said: "I asked Harry to present me with the award because I believe he will beat my goalscoring record.
"It was great to be out there in front of the fans again and it was special for me and my family."