Harry Kane has scored 22 goals in 18 appearances for club and country this season
Kane, 30, has been in prolific form since joining Bayern Munich from Tottenham in August, scoring 19 goals in 15 games for the German champions.
But while he is England and Spurs' record goalscorer, he has yet to win any major honours during his career.
"I'm sure if you spoke to all our guys who have won trophies it changes how you view yourself," Southgate said.
"It isn't going to change what we think he is capable of and isn't going to change his talent.
"But in terms of his mindset and how he will feel I'm sure that's something he will benefit from."
Kane enjoyed a record-breaking career at Tottenham, where he became the club's all-time top scorer with 280 goals in 435 appearances.
His 58 international goals have also made him England's all-time leading scorer - and he was the top scorer at the 2018 World Cup.
Signed for an initial 100m euros (£86.4m) by Bayern, Kane's wait for a first trophy could have ended in his first game for the German club - but his side lost the German Super Cup to RB Leipzig.
He will be hopeful of ending his first campaign in Munich with silverware, however, having joined a team which claimed its 11th successive Bundesliga title last season.
"Any player wants to win trophies, that's what we are all geared to," Southgate said, on the day he announced his England squad for their final two Euro 2024 qualifiers.
"He is of course proud of his individual awards, but if you talk to him he wants to win something with his club or his country. That's how he is driven, like they all are."
Southgate added: "I think clearly it's a missing piece for him at the moment."
Kane's spectacular start to his Bayern career continued on Saturday as he made it back-to-back Bundesliga hat-tricks by scoring a treble against title rivals Borussia Dortmund.
His third hat-trick of the campaign, it took his league tally to 15 goals in 10 games - while he has also scored four goals in as many Champions League group games to help Bayern make a perfect start in Europe.
"His football has always been really good. What's brilliant for him is he's taken on a new challenge, a fresh life if you like," Southgate said.
"He's moved to a big club abroad, which isn't straightforward, but he's adaptable unbelievably well. So I think the confidence he will take from that will be enormous.
"There is a risk we really underestimate what we've got and what we've had for the last few years because any team, any country would love him as their number nine."
-- Courtesy of BBC Sport