The Portuguese appeared to have given up hope of signing the Welshman after admitting that it was 'game over' in his pursuit of the 28-year-old following Tuesday's UEFA Super Cup defeat against Real Madrid.
However, English publication 'The Sun' reports that the former Real boss is privately still optimistic that there is a deal to be done.
The same source claims that Mourinho has told United supremo Ed Woodward to stump up the money required and seal the deal.
Bale has repeatedly insisted that he is happy at the Bernabeu and Real boss Zinedine Zidane has repeatedly rubbished suggestions that the club are looking to sell the player, despite the potential big-money arrival of Kylain Mbappe.
Following the Super Cup, Bale told reporters: “I’m happy to fight for my place. We’re all fighting for our places anyway.
“It’s nothing new in football, it happens at every club, you just carry on like normal.
“Everything has gone as normal like it has every year, there’s been the normal speculation like there is every year.”
Despite this, United are thought to be hopeful that they could tempt both club and player into a deal with a £100m bid and a financial package amounting to somewhere in the region of £500,000 per week.