The Premier League side are believed to be in the market for additions this summer, with manager Brendan Rodgers keen to strengthen his squad.
Ambrose was one of the stars on Nigeria’s recent African Cup of Nations winning campaign, producing performances that caught the eye of the Reds’ scouting team.
Due to this, they are believed to be closely monitoring the 24-year-old, and could send scouts to assess him throughout Celtic’s season run-in.
His agent is thought to be keen to secure his client a big money move, which could press through a deal should the Merseysiders step up their pursuit.
Celtic manager Neil Lennon has confessed that he is ware of interest in his midfielder, but played down speculation that a move could be imminent:
“I know a few Ambrose fans,” he said.
“There is nothing concrete, but I am aware of the interest.”
Rodgers’ current midfield have been criticised for being lightweight in recent weeks, with a number of opponents dominating them physically.
The Northern Irishman sees the combative Ambrose as somebody who could add a steely resilience to the club’s engine room.