The 32-year-old former Chelsea and West Ham United striker joined the Scottish Premiership giants on a free transfer last month.
The Englishman has been working on his fitness for the past few days and could make his debut for the Hoops this afternoon.
Celtic will take on Kilmarnock at Celtic Park in the Scottish Premiership in a few hours’ time, and Cole could be part of the matchday squad.
Former Celtic midfielder and manager Hay believes that the 32-year-old will prove to be a good signing for Ronny Deila’s side.
“Like a lot of people, I am keen to get a look at him because he comes with a decent reputation, and he is at an age where he still could have a fair few years of football in him,” Hay wrote in The Glasgow Evening Times.
“He also comes across as a big, bubbly type of character, and I think players like that always give your dressing room a big life.
“It gives everyone a boost to have another player in anyway, but when it is someone who can come in and be a big player within the dressing room then that is another aspect that brings the team together.”
Former West Ham striker Tony Cottee has predicted Cole to be a success at Celtic and has told The Daily Express: “It might take him another few weeks before the Celtic fans see the real Carlton Cole but I’m sure he’ll be a hit when they do.
“The real Carlton Cole is a handful, a threat at set-pieces with an eye for goal.”
Cole started his professional career at London giants Chelsea, but he failed to play regularly for the first team.
The 32-year-old moved to West Ham from Chelsea in the summer of 2006 and played for them until the end of the 2014-15 season.
The Englishman scored 68 goals in 293 appearances for the Hammers, including one in the 2011-12 Championship playoff final against Blackpool at London’s Wembley Stadium.