Trolls claimed they were going to “shoot and kill” the Easter Road boss while also plotting to “ambush” Lennon in a series of posts on Facebook.
The comments, which also featured numerous sectarian jibes, included one stating that “somebody needs to shoot” Lennon, while another urged “someone give me a weapon”.
Police Scotland officers in Glasgow are now said to be “urgently” investigating the matter.
A spokesman said: “Police Scotland are aware of offensive and threatening comments posted on social media concerning a named individual.”
“Inquiries are currently ongoing into this incident.”
Senior figures at Hibs are said to be aware of the posts and are examining them.
Both the SFA and SPFL did not respond when asked for comment.
Lennon has previously been the target of several safety threats, including being forced to pull out of a Northern Ireland match against Cyprus following death threats from paramilitaries.
In 2011, Royal Mail officers intercepted packages containing bullets sent to Lennon and then-Celtic players Niall McGinn and Paddy McCourt, while in 2012, parcel bombs were addressed to Lennon and two prominent Celtic supporters.
He has also been the subject of threats on Twitter, and was sent a tweet in 2013 reading: “I’ve got a bullet with your name on it.”
The threats came just hours after police also confirmed they had launched a probe into Lennon’s goal celebrations during the match following a number of complaints from Rangers fans.
Authorities received around 30 accusations that Lennon gestured towards the home fans after Hibs’ first goal at Ibrox.
A statement from Rangers fan group Club 1872 claimed Lennon deliberately “attempted to incite trouble,” adding: “Mr Lennon’s actions in the dugout on Saturday – where he made various inflammatory gestures to the supporters sitting directly behind the Hibernian dugout – were not becoming of any football manager, never mind one who likes to play the victim when things do not go his way.”
The statement continued: “Police categorised this match as high risk due to the behaviour of Hibernian supporters at the Scottish Cup final in 2016 when Rangers players were attacked on the pitch.
“Neil Lennon knew that and had a responsibility to act in an appropriate manner, but he shirked that responsibility.”
However Lennon denied the claims post-match, adding: “I’m celebrating a goal, so what way are you supposed to celebrate? If I pick my nose somebody will find something to criticise me for.”
“I’m celebrating a goal, so that’s the end of it.”