Keane has been critical of United's quality as a team as they failed to mount a serious title challenge to Manchester City in the Premier League.
Despite finishing second in the league, United went another season without a trophy after losing to Villarreal in the Europa League final.
England captain Kane, who has reportedly told Spurs of his desire to leave this summer, is also wanted by United's rivals City.
"The only thing I'd hold against Kane, there's a big negative I have towards Kane," Keane told SkyBet.
"I think Kane is an amazing player, every club should be trying to sign him, Manchester United have got to sign him, I don't care if it's £120 million… sign him.
"But I don't think he's a good leader. I never see him digging anyone out at Spurs. I know people might go, 'well that's not his personality' or, 'he might do it in the dressing room'. I don't think he does and that's a big problem.
"Okay, you can lead with your brilliant play. But I think sometimes I want to see another side to Kane, I'd love to see him get hold of somebody."