At the same time, the council is joining forces with Dublin Bus to highlight the issue.
The bus company has agreed to carry ads on its vehicles encouraging bus drivers, taxi drivers and motorists in general to maintain safe distances.
Fine Gael's Eoghan Murphy told the Herald that cyclists often feel nervous when being overtaken by large vehicles.
"Ads on the backs of buses are effective. Dublin Bus came back to us and said they would do it," Mr Murphy said.
However, a similar request to the taxi unions was not so favourably received, he added.
The council's Cycle Forum wrote to Mr Dempsey last year seeking legislation for a "mandatory passing space for vehicles overtaking cyclists in bus lanes".
The Minister replied that "such matters requiring legislative amendment are being considered in the wider context" of two departmental studies.
Cllr Murphy discussed the matter of a 1.5 metre mandatory distance with other members of the council's Cycle Forum this week.
It is understood that while Dublin Bus would provide ad space free of charge, the cost of printing the the posters would be met by the council.
The Dublin Cycling Campaign has already brought the issue to the attention of the Road Safety Authority (RSA), which had been formulating a taxi-driver training scheme.
In a letter, it said: "Cyclists are far from happy with the driving standard of many taxi drivers who claim to be professional drivers."
Among the problems were the "unacceptably high" incidences of taxis overtaking cyclists in an unsafe manner, "that is, far too close and fast within the bus lanes".
"In many jurisdictions, the safe separation distance for a mpv overtaking a rider is set at 1.0-1.5 m minimum width, in law," it pointed out.
The letter added: "Too many taxi drivers overtake riders in the bus lanes, in the main, by 'skimming' past the rider leaving almost no room for error. The driver fails to move out-of-lane to overtake the cyclist."