The Red Devils suffered a disappointing 1-0 loss to Newcastle United at St James’ Park on Sunday afternoon following Matt Ritchie’s second-half winner.
Manchester United are sat in second place but 16 points behind Premier League leaders Manchester City with 11 games to go in the title race.
The 20-time English champions lost 2-1 to Bristol City in the League Cup quarter-finals to prematurely end their defence of the competition.
Manchester United can still finish the season with some silverware ahead of their clash against Spanish side Sevilla in the Champions League round of 16 later this month.
Before the trip to Spain, the Red Devils will host Premier League rivals Huddersfield Town in the FA Cup fifth round at Old Trafford next weekend.
Former Liverpool FC captain Redknapp warned Mourinho that the Red Devils are facing a vital month in the 2017-18 season.
“The next month is pivotal,” Redknapp told the Daily Mail. “Next up are trips to Huddersfield in the FA Cup and Sevilla in Europe.
“Before their return leg against the Spaniards, United host Chelsea and Liverpool and travel to Selhurst Park in the league.
“Mourinho’s superb record speaks for itself. He must demonstrate all those qualities that have made him a world-class coach to ensure that Manchester United end this season with silverware.”
Mourinho led Manchester United to Europa League and League Cup glory last season after the former Chelsea boss took over the reins from Louis van Gaal in May 2016.
The Red Devils finished in sixth place in the Premier League table last term.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news