Luke Shaw has won 30 England caps
Monday's Euro 2024 qualifier against North Macedonia is being played at United's home ground Old Trafford.
Tottenham striker Kane and West Ham midfielder Rice are said to be summer transfer targets for the Red Devils.
"I would love a few of them to join - I am not going to lie," said United defender Shaw, 27.
"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.
"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."
England have been preparing at United's Carrington training base and Shaw posted pictures on social media of him beside Kane, 29, and Rice, 24, with the caption "Carrington tour".
"I was half joking, it is a bit of both," said Shaw. "But if it was down to me, I would take them in a heartbeat.
"Hopefully they like it and can get used to it," he joked.
Striker Marcus Rashford, 25, is set to play on his home ground after scoring a career-best tally of 30 goals for United last season.
"I think when he has had chances this season, you expect him to score," said England boss Gareth Southgate. "Maybe in previous seasons that was a little bit less of a certainty.
"I would say the goal tally is big progress for him. I still think he's better from the left than as a [number] nine but he's definitely made progress as a nine this year and had a bigger impact."
-- Courtesy of BBC Sport