After scoring his 29th goal against Ukraine on Tuesday, the Manchester United man currently lies in sixth place in the goalscoring charts, one goal behind a prestigious fifth spot which includes the likes of Tom Finney, Nat Lofhouse and Shearer himself.
And the BBC pundit, who finished his Three Lion's career with 30 goals in 63 appearances, believes that the 27-year-old United star has the ability to take on the goalscoring legends and become England's greatest-ever striker.
"I hope by the end of this tournament he has smashed my record," Shearer told The Sun.
“If he has then who knows what we will have achieved. He is still just 27, so he could go on and be the number one.
“It is some challenge but that is what Wayne loves. He will certainly be looking forward to the next challenge against Italy on Sunday.”
Shearer also reflected on the England No. 10's lacklustre performance in Donetsk after finishing his suspension and playing his first match of Euro 2012.
“It wasn’t his greatest game and he missed a good chance early on," he explained.
“But he showed his instinct to be in the right place for the goal no matter how the ball got there.
“Great strikers can have poor games but still come out with a goal.
“Now after getting that game under his belt and a goal, I expect us to see a much better Wayne Rooney against Italy.”
England face the Azzurri in the quarter-finals on Sunday, June 24 at 20:45CET.