Man United boss Erik ten Hag is intending to bolster his midfield options this summer with Champions League football returning to Old Trafford next season.
Chelsea star Mount looks set to join the Red Devils this summer with the Blues in need to sell players in order to avoid the Financial Fair Play pinch.
United also hold an interest in Rice but the Hammers skipper is expected to join Arsenal who have been tipped to go on the offensive for him this week.
Carragher insists Mount provides United with an upgrade on Christian Eriksen but Rice would provide Ten Hag with what his midfield requires.
"Mason Mount is an upgrade on Eriksen in terms of where he plays," Carragher told Metro.
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"But going back to Declan Rice, United are one of the few teams at the top who don't play 4-3-3, it's usually two midfielders with Bruno Fernandes.
"Declan Rice and Mason Mount, they are very different players and it shouldn't be one or the other but when you look at what Manchester United have got right now, I actually think Declan Rice would be a better fit for them. A lot more expensive of course.
"I do think he [Mount] is a good player and I like him, I wanted Liverpool to get him. Lots of coaches have picked him and like him. There is already a good player in there but Declan Rice would suit United more than Mason Mount in terms of how they set up."
Meanwhile, former Manchester United captain Gary Neville has revealed his surprise at the club's pursuit of Mount.
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The Red Devils are expected to bid around £55millon for the England international despite Chelsea looking for £70m for their academy graduate.
Neville was quick to praise Mount's attributes but feels his former club has other priorities that need addressing.
"I'm surprised about the Mason Mount one to be honest with you," Neville said. "For the reported £60million fee, he is a very good player, that isn't a question for me.
"But I look at the players United have got already in terms of Bruno Fernandes, Christian Eriksen, Casemiro, Fred and Scott McTominay. Do I think Mason Mount is the player to go in and sort all that out? I'm not quite sure.
"I think you would have been better off going and playing £100million plus on Jude Bellingham or £80-100million on Declan Rice and getting something you think can be the real thing.
"Mason Mount would be a very good addition to Manchester United's squad, but it is a lot of money for a squad addition.
"Does Mount, Casemiro and Bruno work as a midfield three? Spending £60million, you have got to get that jump towards winning the league from third place.
"Manchester United at this moment in time, they haven't got the money and they haven't got the promise of the new ownership to be able to throw the big money at players."