The 29-year-old is set to become a free agent in the summer and he has failed to agree on a new deal with the Premier League club so far.
Manager Neil Lennon wants to strengthen his midfield options and he is keeping tabs on the experienced midfielder’s situation at Burnley.
According to the Mirror, Lennon sees the 29-year-old as a key part of his future plans and therefore the move could be ideal for Brady as well.
SL View: Quality short term addition
The 54-cap Republic of Ireland international could prove to be a quality short term signing for the Scottish giants.
He has shown his quality in the Premier League and he could make an immediate impact at Celtic if he moves to Scotland.
Brady is at the peak of his career right now and he needs to join a club where he can play regular first-team football.
The Hoops will be able to offer him European football and it remains to be seen whether the Scottish outfit can tempt him to move to Parkhead at the end of the season.
Celtic will not be the only club looking to sign Brady once his contract expires and it will be interesting to see if they can fend off the competition and secure the midfielder’s services for the coming seasons.
The 29-year-old can play in multiple positions and his versatility will be an added bonus for the hoops. Brady usually operates as a wide midfielder but he can play in the central midfield as well as a full back.
The hard-working midfielder will not only help Celtic going forward his defensive contribution could be very useful at the other end of the pitch as well.