Every household is to be offered a pair of tickets to events which could include the opening ceremony and 100 metres. Glasgow City Council has sent letters to hundreds of homes eligible for the offer.
The Evening Times understands 5000 tickets will be allocated but residents will not be given a choice of event.
Those who live near the Athletes' Village and velodrome have complained about a concrete and steel cordon. They say their lives have been disrupted by the security measures.
Many of them have now received a letter from the council informing them of a tickets windfall.
A copy of the offer, seen by the Evening Times, said: "A number of tickets will be made available to people living within the areas that will experience particular inconvenience due to arrangements for the Games.
"Each household within these areas is eligible to receive two Games tickets, with sessions available at selected times, dates and venues."
A hotline has been set up for those who want to claim the tickets before July 7.
However, the letter warns: "Due to the high volume of demand and limited availability of tickets across events, we will not be able to take requests for specific sporting sessions."
Calton councillor George Redmond insisted the council must ensure every household is given tickets.
He said: "These free tickets are welcome but there are maybe one or two streets that have been missed out and should be considered.
"We are still trying to work with the council to get additional tickets over and above the initial allocation."
Paul Zealey, head of engagement and legacy at Glasgow 2014, suggested he would support Mr Redmond's call.
Mr Zealey said: "Through our goodwill programmes, Glasgow 2014 and our Games Partners are committed to connecting as many people as possible to the thrilling and positive experience of world-class sport which we're proud to be hosting in Glasgow and Scotland next month.
"We appreciate and value very much the support of those communities closest to the venues and want to ensure they feel part of our sporting celebrations."