It is understood the Black Cats have a deal in place for the 23-year-old, who has been linked with a move back to his native Wales in recent months.
Matthews joined Celtic from Cardiff in July 2011 and has a year remaining on his contract.
He made 42 appearances for the Bhoys last season and picked up a fourth Scottish Premiership winners’ medal in the process.
The news of Sunderland’s bid comes after a busy few days for the Wearside club, who completed a permanent deal for Liverpool defender Sebastian Coates on Wednesday night and then tied up key midfielder Lee Cattermole with a five-year contract extension.
In addition, they are understood to be maintaining interests in Zenit St Petersburg’s Belgium international defender Nicolas Lombaerts and Dutch striker Jeremain Lens, who plays for Dynamo Kiev.