The Red Devils have won five of their eight Premier League matches under Rangnick's leadership, suffering just one defeat, which has moved them into fourth position in the table.
The German is set to step into a two-year consultancy role this summer, with Paris Saint-Germain's Mauricio Pochettino and Ajax's Erik ten Hag thought to be the favourites for the managerial position at the club.
However, according to The Sun, Man United have put their hunt for a new boss on hold, as Rangnick has made a strong impression on club chiefs.
The report claims that the board are watching closely to see how the 63-year-old performs between now and the end of the season, with the German potentially being considered for a long-term role at the helm.
Man United, who have won their last two Premier League matches against Brentford and West Ham United, will be back in action against Middlesbrough in the fourth round of the FA Cup on February 4.