The Premier League giants have signed three new players in Tyrell Malacia, Christian Eriksen and Lisandro Martinez and are still expecting to add Frenkie de Jong this summer.
Erik ten Hag is also hoping to bring in a right-back but that is dependent on Aaron Wan-Bissaka leaving Manchester United in the ongoing window.
Mukiele is one of the players Manchester United have been tracking as a potential solution this summer.
And according to French sports daily L’Equipe, the club are looking to step up on their efforts to sign him later in the window if Wan-Bissaka leaves.
However, PSG have the edge in the race to sign him and are in direct talks with Leipzig over taking Mukiele to the French capital.
PSG are trying to work out an agreement to sign him on loan with an obligatory purchase clause this summer.
The 24-year-old is claimed to be preferring a move to PSG over other clubs at this stage of the summer.
It remains to be seen whether Manchester United look to try and sign Mukiele a little earlier given PSG are now trying to rope him in.