The Scottish Sun reported last month that Blackburn are looking at making a move for the 29-year-old, who is a very versatile player and can play as a centre-back, a left-back or a central midfielder.
Earlier this week, The Daily Mail reported that Mulgrew is a summer target for Premier League outfit West Bromwich Albion.
Johnson has confirmed that Championships side Blackburn are keeping tabs on the Scotland international.
The Rovers head of recruitment watched the former Aberdeen man in action for Celtic against Dundee at Celtic Park in the Scottish Premiership on Wednesday evening.
“I know he is out of contract like a lot of other players,” Johnson told The Scottish Sun about Mulgrew. “We have got a pamphlet that is about 300 or 400 pages long with players who are out of contract.
“We just sit down and go through them to see if they fit our remit, our wage structure, if their personality fits into the club.
“I have been coming up watching games all season, not just Celtic. I will sit down with the gaffer and we will discuss players for the summer.”
Celtic are at the top of the Scottish Premiership table at the moment with 63 points from 28 matches, four points clear of second-placed Aberdeen.
Ronny Deila’s side will take on Greenock Morton at Celtic Park in the quarter-finals of the Scottish Cup on Sunday afternoon.