Moves for either Antoine Griezmann or Gareth Bale remain an option, but it is now claimed Lemar has shot to the top of Mourinho's wanted list.
Mourinho made three major signings this summer in Victor Lindelof, Romelu Lukaku and Nemanja Matic - but missed out on adding the wide player he craved to his attacking armoury, failing to get a deal for Inter Milan's Ivan Perisic over the line.
Lemar's advisors are aware of interest from Mourinho as the chase for the Frenchman hots up.
He was subbed at half-time as Monaco were thumped 4-0 by Nice on Saturday night with Paris Saint-Germain look set to run away with the French title this term.
Any move for the France worldwide will likely top the £89m club-record fee shelled out for Paul Pogba past year.
The Chilean striker's contract runs out at the end of this campaign and he is unlikely to sign an extension.
The Gunners had a British record £92m bid accepted by the Ligue 1 champions, only for the 21-year-old to reject the move to the Emirates, with reports suggesting he didn't want to make such a big move with so little time, and while on global duty.