It means Carrick will be in charge for a third successive game, having already led the team to a 2-0 win at Villarreal in the Champions League and a 1-1 draw away to Premier League leader Chelsea.
United's hopes of upgrading from caretaker Carrick to interim boss Rangnick in time for the Old Trafford clash with the Gunners have been hit by red tape.
As the team went through their preparations for the game at the Carrington training base, Carrick told United's players he would be giving out the orders once more.
In a statement on their official website, United said: "Ralf Rangnick was announced as interim manager on Monday but, while the club follows the regulatory process around a work visa, Carrick will continue at the helm.
Find out which international job he once interviewed for... 👀#MUFC— Manchester United (@ManUtd) November 30, 2021
"Michael has communicated this to players and staff at Carrington in a busy week for the club."
United announced on Monday an agreement had been reached with Rangnick over a relationship that is set to span two and a half years.
Initially, Rangnick will take charge until the end of the season before moving into a consultancy role.
Rangnick has previously shaped German clubs such as Hoffenheim and RB Leipzig from top to bottom, with his future advisory position at Old Trafford suggesting United are planning an overhaul of their football operations, which have long appeared outdated compared to their rivals.
He has been working most recently at Lokomotiv Moscow, but the 63-year-old is now set to step into the highest-profile position of his career.
Carrick has been at the helm since Ole Gunnar Solskjaer's dismissal on November 21, which followed a dismal run of form that saw United lose five of seven Premier League games.
It sits eighth in the table, five points behind fifth-placed Arsenal, having won just five of 13 domestic league games this season and with a goal difference of minus one.
United is at home again on Monday (AEDT) when it hosts Crystal Palace, seemingly a more realistic target for Rangnick's first game.