The Red Devils have failed to win their last four matches in all competitions, most notably losing 3-1 against West Ham at the London Stadium on Saturday.
That result left Mourinho's men nine points behind local rivals Manchester City in the Premier League table, which followed a third-round Carabao Cup exit at the hands of Championship side Derby County last week.
United's performance in the Champions League against Valencia was similarly uninspired, as they failed to carve out any clear goalscoring opportunities while also looking uncertain in defensive areas.
However, while United legend Paul Scholes has claimed Mourinho is 'embarrassing' the club and is surprised the Portuguese has not been sacked, former Red Devils striker Dublin offered his backing to the under-fire manager.
Dublin told BBC Sport: "I was at Old Trafford for BBC Radio 5 Live and although the crowd was clearly unhappy I did not get a sense they were anti-Jose.
"I was asked after the game if I still think he is the right fit - and I think he still could be.
"However, something dramatic has to be said or done for that to happen, because United need to be in harmony and, right now, that is definitely not the case."
Mourinho must now galvanise his squad for a home clash against Newcastle on Saturday, which is a must-win game for the 13-time Premier League champions if they are to re-invigorate any sort of title challenge.
Dublin went on to suggest that the players should be shouldering more of the blame for the team's recent woes on the pitch and giving their all in the famous red shirt rather than sulking, as he added: "Yes, it is Mourinho who has to drive any improvement. He is the manager and it is down to him to get the players to do exactly what he wants them to do on the pitch.
"But the United players have got to take responsibility too.
"At the moment I don't see enough of them really digging deep and saying, 'right I am going to take this game by the scruff of the neck, and make something happen'.
"I have not seen it for a long time, and it is very frustrating."