The former Sampdoria midfielder is currently on a season-long loan at Atletico Madrid, but has found playing time difficult to come by and is expected to head elsewhere this month.
There is interest in Serie A where his stock remains high, but according to Italian outlet Tutto Mercato, Monaco are putting in a strong push to sign Torreira.
The Ligue 1 side are offering Torreira a three-year deal to swap the Emirates Stadium for the Stade Louis II in the sun-drenched principality.
It is claimed that Monaco’s proposal appears to have found favour with both Arsenal and the player.
The coming days are expected to be crucial to finding out whether the deal can be pushed over the line.
Monaco are continuing discussions to try to put the move in place and snap up Torreira before the transfer window shuts.
The principality side currently sit in fourth spot in the Ligue 1 standings.
