To be honest I don't think Manchester United would have much chance to sign Harry Kane in case of Real Madrid's joining the race to sign him.
Because Real Madrid are in a much more favourable position for Kane if he doesn't make leaving England a problem. It is up to that actually. I wonder if Kane is okay to leave the country as well if he wants to leave Tottenham. If his answer is yes then Real Madrid and maybe Bayern Munich also would be his best options. Otherwise Manchester United can have the upper hand on these teams and bring him to the team. I don't know if Levy can keep Kane anymore if Tottenham have another bad season without titles.
First, Manchester United can lure him with money, besides at the moment the Red Devils are doing quite well in the league. Secondly, if he does go to Real Madrid or Bayern, what will he be there? In Spain there is a high probability of getting lost among other high-quality players and then it will be a bad transition for Kane, which he will soon start regretting
It surprises me that everyone is talking about trophies. So if he chooses Real Madrid, he will surely get some trophies? It looks like helping the weak... Going to Manchester United or Bayern he would have the same chances of winning a trophy with the team. That's what the English league is famous for, every team there can win something, so MU and Tottenham have the same chances in this regard