I don't know whether Manchester United can sign Mbappe in case there is an opportunity this summer. Because they wouldn't be the only team interested in him. Real Madrid would surely get into this race right away as they have been monitoring Mbappe for a long while. They are also in need of a striker now as this need has got bigger as of Benzema's departure.
Manchester United have got way richer than before certainly. But as long as Real Madrid present the same conditions on his contract I don't think Mbappe's choice would be the Red Devils. Because they haven't been as successful as Real Madrid in the recent years before anything else. Mbappe must be hungry for winning a Champions League title.
Personally, I doubt Man United to sign Mbappe. Man United may have the money to sign him, but Mbappe should prioritize Real Madrid over Man United. Real Madrid is his dream team and Real Madrid offers a bigger chance to win any top trophies. Specifically, Real Madrid is a perfect team for Mbappe if he dreams to win UCL trophy. So, he should assume to have a better career in Real Madrid. Well, because of these reasons, I'm very doubtful if Man United can change Mbappe's mind. Since the chance is likely very small, I think Man United is better to look for another striker.
In addition, Real Madrid constantly monitors Mbappe's situation in PSG. If Mbappe has the intention to leave PSG, Real Madrid must be the first team to know it. So, Real Madrid is likely leading the competition to sign Mbappe.