According to Italian source La Gazzetta dello Sport, the 49-year-old is ready to relinquish his post at the Italian club following criticism from fans and has attracted interest from both United and Barça.
With Roma having garnered just four points from a possible nine this season, Monchi has found himself under immense pressure and could possibly step down from his role.
The former goalkeeper has shouldered the blame for the club's poor start due to his questionable recruitment over the summer. Supporters are understood to be very disappointed with the transfer policy that saw the likes of Alisson Becker, Radja Nainggolan and Kevin Strootman depart.
Fans are concerned that the sporting director's plans to sell players for large sums and replace them with cheaper acquisitions are hurting the club. And, admittedly, it has shown in the results.
I Giallorossi sold Alisson to Liverpool for £67m and replaced him with Robin Olsen, who cost just £7m, while Strootman's sale to Marseille left them on the end of a lot of stick, given the resulting lack of depth in midfield.
Monchi is now understood to be considering his future, with United and Barca both taking up an interest.
The Red Devils, in particular, want to bring in someone to take some of the load off of Ed Woodward, with their summer transfer business largely considered to be a failure as the club missed out on more than a handful of targets.
Jose Mourinho is also thought to be extremely disappointed with the backing he received during the transfer window and would likely be pleased if someone like Monchi stepped in to oversee transfers.