The League One club yesterday notified Police Scotland of "deeply offensive and threatening comments" that have been made on the Follow Follow internet forum for the club's supporters.
The post allegedly mentioned the possibility of "torching" the director's car or house.
The director, who is not being named by The Herald at the club's request, was on business in London at the time while his wife and child were at home in Scotland.
A Rangers spokesman said the comments put the man and his family in a state of fear and alarm.
He said: "Police are looking at this and they are conscious that there is a particularly toxic atmosphere on the web at the moment."
In a statement, the club said: "Rangers FC will not tolerate this and intend to take an extremely robust approach to this sort of behaviour. The board finds it inexplicable that so-called supporters of the club are bringing Rangers into disrepute by this appalling behaviour. These people are not welcome at Ibrox.
"The board is also aware that certain individuals are holding meetings and inciting fans to unruly behaviour. This has also been reported to the police."
It is not the first time police have had to look at allegations of this nature.
Last November a 37-year-old was arrested over alleged offensive online material surrounding Rangers' financial affairs.
In December, a 40-year-old fan and podcaster was arrested for allegedly making offensive remarks.
Police have been proactive in targeting crime online and on social media outlets such as Facebook.
Earlier this month, the Rangers board announced it had avoided calling a general meeting over calls for the removal of directors.
Disgruntled shareholders had called for the meeting, where they wanted chief executive Craig Mather, financial director Brian Stockbridge and board member Bryan Smart removed.
The board had only days to sort out the dispute but later confirmed that the general meeting requisition notice had been withdrawn on the condition that Rangers International Football Club PLC holds its first annual general meeting no later than October 31.
All current directors of Rangers International will have to resign and put themselves forward for re-election by shareholders at the AGM.