For 20 years, a Republican parade to honour Connolly, who was executed for his part in the Easter Rising of 1916, was held in the city.
The march, which was regularly hit by sectarian disorder and was banned twice in the 1990s, became the largest Republican parade in Scotland and attracted 3000 participants.
It was shelved in 2007 with its organiser, The James Connolly Society, citing the changing political situation in Northern Ireland and a desire to curb sectarianism in Scotland.
But a separate group, The Celtic Commemoration Committee (CCC), has now applied to hold a march in memory of Connolly on June 9.
William McDowall, who submitted the application, told police and council planners the march would be a “family event”. But Assistant Chief Constable Bill Skelly said it was inevitable that the application would be linked with previous marches in memory of Connolly by sections of the public.
He said: “Terrorist activity continues to feature in Northern Ireland and the CCC openly state their support for a united Ireland, therefore their views and opinions are directly linked to Irish republican sentiments.
“It can be reasonably foreseen a CCC event will attract some level of opposition. If the event goes ahead, there may be some disruption and it would be wrong to totally discount the possibility of disorder.”
Police have suggested that a series of conditions, including bans of militaristic uniforms, inflammatory banners and a 45-minute time limit on a rally, are put in place.
Around 750 participants and two bands are expected to attend any Edinburgh event.
Police are braced for the possibility of Loyalist groups applying to stage their own rally in opposition.
City centre councillor Joanna Mowat said: “I wish they wouldn’t march but we live in a country with free speech and that’s too precious to say they can’t march. If it can take place in a peaceful fashion there must be a presumption that it should go ahead.”
Her fellow ward councillor, Alasdair Rankin, added: “As long as it remains peaceful, marching is perfectly acceptable in a democracy.”
The route which the march would take has not been decided. Mr McDowall originally applied to march from Grassmarket to The Meadows, where a rally would take place, but neither the council nor police approved the route.
The application is due to discussed by the licensing sub-committee on Friday.
JAMES Connolly was born on June 5, 1868, at 107 Cowgate to Irish parents.
By the time he began his first job at 12 he was aware of political unrest in Ireland.
At 14 he had to “take the shilling or starve” and from 1882 to 1889 was stationed with the King’s Liverpool Regiment in Ireland.
Connolly entered politics upon his return to Edinburgh, becoming an activist in the Scottish Socialist Federation, before emigrating to Ireland in 1896, where he set up the Irish Republican Socialist Party. From 1914 he looked at ways to disrupt the British war machine and he joined Sinn Fein.
He was in command of Dublin’s GPO at the Easter Rising of 1916. When the Irish Citizen Army surrendered, Connolly was captured by the British. He was execu-ted on May 12.