According to 'Sky Sports', Middlesbrough are set to take the player on loan with an option to buy on a permanent deal at the end of the season.
However, it is also understood that Millwall will want to have a replacement lined up before Saville can leave the club, with Friday's loan deadline fast approaching.
Middlesbrough made an effort to bring in both Saville and Jed Wallace on Deadline Day, though several offers were rejected by the 'Lions'.
Saville, 25, came through Chelsea's academy before joining Wolves, who he left for Millwall in a deal worth £350,000 last summer.
The Northern Ireland international was also part of his country's side that lost to Switzerland in the World Cup play-offs.