Carrick remains unbeaten as manager since succeeding Ruben Amorim after his January 5 dismissal.
The Geordie has been contracted to the end of the season as the board consider a long-term appointment.
And The Sun says United's players want to work with Carrick over the coming seasons.
Senior players Bruno Fernandes, Harry Maguire and Luke Shaw have all raved to family and friends about Carrick's management and tactical nous.
A source said: “Michael has come in and done great because he has been a breath of fresh air.
“Some of the players expected that, because playing for Amorim was like playing chess. You had to follow his rules to the tee, whereas Michael has told them to express themselves.
“What they are really impressed with is his knowledge of the game. He knows about players in other leagues and clearly spends an awful lot of time thinking about football and following it.
“The players know it’s not their decision, but the majority would be happy if he got the job permanently.”
United can jump to third on the Premier League table with a home win over Crystal Palace today.