The call, which lasted for almost an hour, was aired on the FM104 Phoneshow last night.
The individual phoned the station in a highly distressed state, claiming that he was on the verge of jumping off Loughlinstown Bridge in south Dublin.
Gardai managed to persuade Jay to climb down in the early hours of this morning.
Independent TD Finian McGrath today called on the Broadcasting Authority of Ireland (BAI) to investigate the decision by the station to air the phone call.
His family was alerted to the call, which prompted his father to phone in himself.
The station went off air for 10 minutes, during which the man's father is believed to have urged him not to follow through with his threat.
Listeners were horrified when they heard the man angrily shout at gardai: "Get off the bridge".
Presenter Jeremy Dixon admitted during the controversial show that he was "not qualified to deal with this".
When the programme returned to the air, presenter Jeremy Dixon told listeners: "The line has gone dead."
He announced that the show would be extended in case Jay wanted to call back and added that he was "cut up about this".
"It is obvious he has issues with a number of people, including gardai and the hse," Dixon told listeners.
FM104 and Mr Dixon were unavailable for comment this morning.
Dublin TD Finian McGrath called the station's actions as "grossly irresponsible".
"I am highly concerned that this radio station would broadcast a call from a man who was in such a vulnerable state.
"Suicide is such an emotional topic for people at the moment and the station went too far by broadcasting that call. I am further demanding that the BAI launches an investigation into the decision."