The Red Devils full-back has been using the time spent with England to speak to both Rice and Tottenham striker Harry Kane about the merits of moving to the north-west.
He was helped by the Three Lions using Carrington as their base before thrashing North Macedonia at Old Trafford on Monday night.
And he said he would have them as club team-mates “in a heartbeat”.
“It is not normally discussed but, because of the time of the season and what is going on, there is a lot of talk, a lot of banter,” Shaw said.
“I would love a few of them to join United – I am not going to lie. I am speaking to them, telling them how good Manchester is because I know their qualities.
“I know they are world-class players and they would be a massive help to us but I can only say so much. It is up to the club and whatever they decide, so we have to go from there.”
Shaw also joked that “hopefully they like it and can get used to it” at United, who have a long-standing interest in Kane and Rice.
They are, however, understood to view deals for the duo as highly unlikely.
Rice remains much more likely to wind up at Arsenal and now international duty is done, movement of one form or another is expected to come quickly.
The Gunners have already had one bid rejected but are believed to be considering a second in the region of £100m.
The 24-year-old midfielder meanwhile started both of England’s European Championship qualifying games, the other being a 4-0 success at Malta on Friday, playing the full 90 on the Mediterranean island before being substituted just under an hour into the Macedonian romp.
The results leave England top of group C with four wins from four.
There was almost a glorious international debut for the Hammers’ Callum Marshall as Northern Ireland lost 1-0 away to Denmark.
The 18-year-old had put the ball in the net in second-half stoppage time only for a VAR check, that took five minutes to conclude, to rule it out for offside.
Manager Michael O’Neill had draped an arm round the shoulder of the West Ham youngster at the final whistle but said the finish itself, an instinctive flick to turn Jonny Evans’ header goalwards, showed the talent he has.
The boss said: “What he did show with his finish is that’s not something you coach, that’s just an instinctive player who had the ability to steer the ball towards the net in that situation.
“It’s nice to have that but it doesn’t mean we can rely on him or we can expect it every time he comes on the pitch.
“But we were delighted with the contribution of him and all the subs.”
Skipper Evans also had sympathy for the teen striker.
He said: “I actually spoke to someone who was at the game and I think the fans were quite buoyant after the game.
“The emotion of scoring the goal, alright it got taken away, but it gave everyone that feeling.
“For young Callum, scoring the goal, to get that feeling hopefully it makes him more hungry and for us as a team it shows we can go and compete and score goals away from home.
“Alright it was offside but they were very fine margins.”
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