According to Goal's Kris Voakes, PSG could move for Mourinho but United are keen to sign him to a new deal less than a year and a half into his three-year contract at Old Trafford.
The Portuguese and his advisers will be presented with terms worth £250,000 per week "in the next few weeks" as United look to tie Mourinho down for the long term, Voakes noted.
Despite being successful in his previous roles, Mourinho has only once completed more than three full seasons during a single spell in charge of a club.
The 54-year-old recently revealed to TF1's Telefoot programme that he has ambitions in management beyond United (via Voakes): "The only thing I can say is that I am still a coach with worries, with ambitions and with the desire to do new things. And I don't believe...no, I'm sure that I won't end my career at United."
He also praised PSG as "something special" in the same interview, sparking rumours he could move to the Parc des Princes.
But Mourinho later clarified he was "not leaving for PSG," and United look set to try to tie him down to a longer deal, per Voakes.
United began the 2017-18 season in fine fashion but have dropped five points behind Premier League leaders Manchester City after a 0-0 draw with Liverpool and the recent shock 2-1 defeat to Huddersfield Town.
They are ahead of the third-place Tottenham Hotspur on goal difference and arguably face a must-win clash at home to the north London outfit in the league on Saturday.