Rooney's present deal expires at the end of next season, and with the 28-year-old 41 adrift of matching Charlton's overall 249-goal Red Devils career, realistically he will not pass it until the 2015-16 campaign.
That would evidently require Rooney to accept the lucrative new contract offer United are prepared to make.
"It is a great aim for myself to try and get Sir Bobby's record, both at club and international level," Rooney told BBC's Football Focus ahead of Sunday's Barclays Premier League encounter with rock-bottom Fulham.
"It is something which I would like to do. If I can do that I would be really proud because it has stood for a long time."
Rooney is 11 goals behind Charlton's England record of 49, having scored in the vital World Cup qualifying wins over Montenegro and Poland at Wembley in October.
United have struggled badly during David Moyes' debut campaign as manager, suffering their eighth Premier League defeat of the campaign at Stoke last Saturday.
It was a loss that left the Red Devils seven points adrift of fourth-placed Liverpool, which both Rooney and Moyes admit is their only realistic aim now.
"We know it is going to be extremely difficult to win the title now," said Rooney.
"There are a lot of points in between ourselves and a few teams.
"Realistically, the top four has got to be our aim. Although, as a club, we always want to win the league title maybe it might be a bit too far."
Moyes also expressed a similar view in United Review: "We must continue to stick together and try to produce enough wins to make sure we are in the Champions League next year.
"We will continue to do everything we possibly can to make that happen."
Moyes' own position is yet to come under threat, and Rooney does not believe the Scot bears the responsibility for results anyway.
"David Moyes is a fantastic manager," he said. "I worked with him at Everton and now I have worked with him for a number of months at Manchester United.
"He is doing everything he can do but it is the players - myself and the rest of the team - who have to do better."