Which one has a greater rivalry: Manchester United vs Liverpool or Manchester United vs Manchester City?
A good comparison would be to say that Liverpool vs Everton is a bit like Atletico v Real Madrid or Dortmund v Schalke 04, where as Man Utd v Liverpool is more like Barcelona v Real Madrid, or Bayern Muenchen v Dortmund.
Manchester United vs Liverpool is definitely more heated. The local rivalries in Liverpool and Manchester do cause some trouble, but in Liverpool you'll see fans of both clubs drinking together after the games, in Manchester less often.
Mancs and Scousers are bitter rivals outside of football, basically the remains of an industrial revolution pissing contest about which city is superior. In football, Man U and Liverpool barely exist to each other. No one's transferred between the clubs since 1964. Before Michael Owen left Man U in 2012, I'm pretty sure the last player to have proper spells at both clubs was Paul Ince, that was 16 years ago. Having both clubs on your CV is an anomaly at best and a cause for suspicion at worst.
Manchester United and Liverpool have also battled it out for more titles than any of us have had hot dinners. Man City, on the other hand, have only been able to contend with their rivals for the last few years of the Premier League era and you have to look back to the 1970s for the last time the two clubs were on the same level.
Despite City's new found success, United's rivalry with Liverpool is greater. In truth United's rivalry with Leeds is greater than its rivalry with City (though it's been somewhat quelled by a lack of games due to Leeds' prolonged difficulties.
A few decades ago, when travel was more expensive, it wasn't uncommon for people in Manchester to watch United one weekend and City the next; not a breeding ground for fierce animosity. Although that's changed most United fans still view City as an irritant, rather than an enemy.
It depends quite a lot on the fan's age, and also where the fan in question comes from.
Certainly for people who support Man Utd and are from Manchester through and through, Man City has always been a huge fixture. Likewise for Liverpool fans in the Merseyside and North Wales area Everton are their most important rivals - For those fans, they are more likely to know other Man City, Everton fans in their friendship groups and families growing up, hence the banter and fierce one-upmanship involved in day to day life. To them Man Utd v Liverpool is huge (and also an honourable mention to Man Utd v Leeds derbies due to the Lancs/Yorks regional rivalry that goes beyond football and sport) - however largely in terms of the competitive rivalry. You won't get many families in their heartland respective regions where brothers, and uncles support each others teams, but Man Utd and Liverpool fans would have that rivalry most definitely with Man City and Everton fans respectively.
What differs though - and the answers already on here confirm this - is that both Liverpool and Man Utd have more global fanbases (and widespread fanbases within britain outside of their regions too). For their 'global fanbases' Man Utd v Liverpool carries more weight. I'm sure City have more fans in Indonesia or Pakistan nowadays, but not to the same extent as the other two. Certainly Man Utd v Liverpool is the big money 'global fixture' for the league in terms of worldwide interest and television revenue.