Rhodes joined Middlesbrough from Championship rivals Blackburn Rovers on transfer deadline day for an initial fee of £9 million, as reported by BBC Sport.
Former Newcastle striker Shearer believes that the Scotland international could score up to 15 goals for Boro this season.
“You can also look at Middlesbrough in the Championship who paid somewhere around £9 million for Jordan Rhodes,” the former England international wrote on Coral.co.uk.
“They want a slice of the pie next season. It’s a lot of money to spend on a Championship player but he will guarantee you goals at that level.
“When you look at his goalscoring record, it would suggest he could score 10-15 goals from now until the end of the season which would probably be enough to get them promoted.”
Rhodes had been on the books of Blackburn since 2012 and is rated as one of the best strikers in the Championship.
The 25-year-old scored nine goals and created 14 chances in 24 Championship games for Rovers this season.
The striker had a pass accuracy of 64% and a shot accuracy of 50%, and won 64 headers, one dribble and six tackles.
During the 2014-15 campaign, Rhodes scored 21 goals and created 38 chances in 45 Championship games for Blackburn.
Middlesbrough are second in the Championship table at the moment with 55 points from 27 matches, just a point behind leaders Hull City with a game in hand.