Rooney, England's record goalscorer, has nine goals in 20 games in all competitions this season, but has lost much of the explosiveness that marked him out as a special talent when he burst on to the scene as a 16-year-old for Everton.
He has scored twice in 11 Premier League games for Manchester United this term, well behind the league's top scorer Jamie Vardy (12 goals) of Leicester City, and Kane, who has five in his last three league games.
"I want to prove myself. I feel that since Roy has been in charge, my international form has been great," Rooney told reporters at a pre-match news conference.
"I have scored a lot of goals and I hope that will continue."
The imminent returns of Daniel Sturridge, Theo Walcott and Danny Welbeck will also increase Hodgson's selection headache for next year's European Championship finals in France.
Welbeck has a knack of finding the back of the net for England and scored six in his last seven games for his country before injuring his knee.
Sturridge, who has played only three times this season, has scored two league goals.
Walcott, expected back after the international break, is finding his feet as a central striker, and like Vardy can also play out wide.
"It's certainly the best group of strikers since I've been involved in the squad," Rooney said.
"It's a huge challenge to keep your place in the team and that's what I want to do."
Hodgson also backed Barkley to thrive playing behind the striker as England gear-up to face a higher quality of opposition than they encountered in their Euro 2016 qualifying group.
"We're going to find ourselves in a situation now where games should be more cut and thrust," Hodgson said.
"Counter-attacking football has not really been much of an option (in the qualifiers) because the opponents have stayed so deep.
"We think the type of players we have in the team are well suited to counter-attacking football and Barkley, of course, is one of those."
(Reporting by Shravanth Vijayakumar in Bengaluru. Editing by Patrick Johnston)
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