Rooney suffered a dead leg in last weekend's 2-1 win at Crystal Palace and Shaw sustained a facial injury, the latest in a succession of fitness problems to plague the 19-year-old in his first season at Old Trafford.
United manager Louis van Gaal said he did not want Shaw to play for the England under-21 team at this year's European Championship.
"When you are not playing for Man United you cannot play for other teams," Van Gaal told reporters on Friday.
"He needs the rest because he has over-crossed the line. He is 19. We have to protect him. I hope the coach of the under-21 team understands that."
United are fourth in the table, six points ahead of fifth-placed Liverpool, and with a vastly superior goal difference they should secure a place in the Champions League qualifying round even if they lose to Arsenal and Hull City in their last two games.
Victory over the Gunners would lift United one point above their opponents into third but Arsenal have a game in hand.
Influential midfielder Michael Carrick will also miss Sunday's game but striker Robin van Persie is available to face his former club along with Argentine defender Marcos Rojo.
(Reporting by Ed Osmond, editing by Tony Jimenez)
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