This comes after Adam Idah‘s departure to Swansea and failed attempts to sign Kasper Dolberg and Kelechi Iheanacho.
With the Europa League squad deadline set for 11pm, the Scottish champions are eager to bolster their attacking options.
Bamford, who recently ended his seven-year stint at Leeds by mutual consent, has been contacted by Celtic.
The 31-year-old striker helped Leeds to two Championship titles, scoring 60 goals in 205 appearances.
However, last season saw him restricted to 17 Championship appearances without a single start.
Leeds manager Daniel Farke confirmed last month that Bamford had no future at the club.
Bamford, who earned his sole England cap in September 2021, has also been linked with a move to Wrexham.
It is yet to be seen if a deal with Celtic can be completed before the Europa League deadline.
Celtic’s pursuit of Bamford comes after they missed out on signing David Fofana, who chose a move to Charlton instead.
Although the club secured winger Sebastian Tounekti in a £5million deal from Hammarby, they pulled the plug on a loan move for Burnley’s Manuel Benson after Birmingham City ended their interest in winger Yang Hyun-jun.
Despite the uncertainty surrounding a potential deal with Bamford, Celtic are determined to strengthen their squad for the upcoming Europa League campaign.
Restricted to free agents and out of contract forwards, Celtic are also interested in Nigerian international Emmanuel Dennis.
The 27-year-old left Nottingham Forest by mutual consent at the end of the winger after three years at the City Ground.