Simon Jones of MailOnline (h/t Metro) reported United sent a scouting party to watch Max against Hertha Berlin in the Bundesliga, with Chelsea, Tottenham Hotspur and Everton interested in the defender.
The 24-year-old has been hugely impressive this season, and his performances have seen his reputation explode in recent months.
Max has the ability to play at wing-back or midfield, and his raiding style has seen him gain seven assists in 15 Bundesliga appearances this term.
The Viersen-born defender is closing in on a call-up for Germany, and he has been one of the breakout stars of the current campaign.
Augsburg are in seventh after losing just four games this season, and Max has been a major reason for the team's form.
The full-back gained a silver medal for Germany at the 2016 Olympic Games in Brazil, and he has dramatically improved since that success.
Mourinho could also turn to Fulham teenager Sessegnon, with the prospect rated at £50 million.
Mike McGrath of The Sun (h/t the Mirror) reported the Cottagers are ready to sell the 17-year-old, who is one of the hottest talents outside of the Premier League in England. Fulham are reviewing left-back targets, with Sessegnon preparing for the step up to a major club. Spurs will rival the Red Devils for the player's services if he becomes available in January.
The English talent is more than just a full-back, and he has an eye for goal as he attacks with vigour from wide positions.
Sessegnon could develop as the best wing-back in the country in seasons ahead, and any prospective buyer should take a punt on his services as soon as possible.
Here is the player in action:
United have plenty of cover at left-back, allowing the youngster to arrive with less pressure to perform in his earliest days.
Max would be the cheaper option, but there is always a premium to pay for outstanding English talent.
Sessegnon certainly falls into this bracket, and he possesses similarities to Gareth Bale at the same age.
The huge advancement in transfer fees over the past year could see the youngster turn into a good value, long-term signing, especially if his progress continues at the current rate.
The defender has already outgrown his surroundings in the capital city, and he could pair with Marcus Rashford on the left to form one of the most exciting combinations in European football.