The 28-year-old scored the 41st goal of his international career in his side's 1-0 win over Norway on Wednesday and now trails the England legend by just eight goals.
Rooney told reporters: "I'd be a fool if I didn't look at Sir Bobby Charlton's record and feel capable of passing him. It's in my sights. I want to do it. Hopefully one day I can. I speak to him a lot at Manchester United - but that's not something we really talk about!
"Obviously it would be a great achievement by myself, and I'd be delighted to do that. Both records have stood for a long time, and a lot of players have attempted to break them.
"Now they're in touching distance, the two of them. It's something I'd love to do. I feel I'm capable of doing it. Hopefully one day I'll hold the two records. I'm confident I can."
England take on Switzerland in their opening match of their Euro 2016 qualifying campaign later today.