United have spent in excess of £100million in recent weeks to recruit Jadon Sancho and Raphael Varane from Borussia Dortmund and Real Madrid respectively.
But Neville believes his former club should not rest on their laurels and must now lure Kane north from Tottenham before the transfer window closes on August 31.
Kane, who made his first appearance of the season off the bench in the win over Wolves on Sunday, sparked frenzied speculation around his future when reports emerged in May that he wants to leave Spurs.
Manchester City have been strongly linked, but Neville thinks it is United who need to make a statement signing in order to keep pace with their rivals and back boss Ole Gunnar Solskjaer.
He said on Sky Sports' Monday Night Football: "Chelsea, we've seen what they've done with [£97m signing Romelu] Lukaku. We've seen that [Virgil] Van Dijk has come back [from injury] for Liverpool and we know that Man City are in for [Jack] Grealish and Kane. If they get Kane, I think there's a massive problem for United.
"I'm struggling to understand why Manchester United aren't going that extra step this transfer window.
"Sancho for 70-odd million was last year's money and I think with regards to Manchester United they've always gone after the best player in the Premier League, the best English player in the Premier League, or British player historically.
"[Anthony] Martial, [Daniel] James, [Jesse] Lingard – you're talking about potentially offloading a lot of wages there, potentially getting some transfer fees in.
"Why are they not going in for Harry Kane this week?
"I think if Harry Kane entered Manchester United you could get up to 90-95 points.
"Without him, they are still going to have to replace [Edinson] Cavani and Martial next summer – Martial is not going to do it, Cavani has only got one year left.
"They are going to have to buy a centre-forward at some point – why not get the man now and give Ole that impetus?"