The future of the 23-year-old has come under close scrutiny over the past week after Rangers submitted two bids for the midfielder, two days before the teams met in the Petrofac Training Cup last weekend.
Allan has been the subject of abusive tweets and Police Scotland said on Twitter on Wednesday afternoon: “We are aware of offensive comments made online towards Hibernian footballer Scott Allan. Police officer enquiries are at an early stage.”
Allan did not make the starting line-up for the Easter Road tie last Saturday, although he was introduced from the bench during the second half of the 6-2 defeat.
Hibs expressed their displeasure on Tuesday at the two approaches for the former Dundee United and West Brom player insisting he would not be sold to their Scottish Championship rivals.
In a club statement they described the two bids of £175,000 and £225,000 as “unwelcome and uninvited”.
They added: “Hibernian FC has refused both bids because the club has no wish to transfer a valuable player to strengthen a major rival in the battle to win the Championship and gain promotion to the Premiership.
“Hibernian’s position has not changed. The player will not be sold to Rangers.”
Allan started his career at Tannadice before moving south in 2012 and spent time on loan with Portsmouth, Milton Keynes and Birmingham during his spell at the Hawthorns.
Rangers were interested in signing the Scotland U21 international last summer but he chose to join the Easter Road club having previously said he would not be keen on playing in Glasgow prior to his move to West Brom.