Sunderland defender Matthews has hardly featured this season.
The right-back moved to Wearside in the summer transfer window, signing from Celtic, but he has been handed few opportunities at the Stadium of Light.
He played 36 minutes as a substitute against Leicester on the first day of the season, but has not played in the Premier League since then.
The 24-year-old started and completed the League Cup second round win over Exeter, but the rest of his season has been spent in the Under-21 set-up.
It is clear he is not in favour under Sam Allardyce, and Matthews move to Sunderland has ruined his chances of heading to the European Championship this summer with Wales.
With just weeks of the season left, Matthews is looking for a move away from Wearside, and has gratefully been targeted by Bristol City.
The Robins are struggling in the Championship, sitting in 20th place with 37 points from 35 games, and have been looking to stabilise their defence recently.
Matthews has travelled to Bristol to start training with the side today, with a loan deal expected to be confirmed on Monday afternoon.