More than a century after a transit line into downtown from the east end was first proposed, Toronto's politicians are staring down fast-approaching deadlines to agree on funding that will move the long-stalled project forward.

If councillors don't approve spending on an accelerated environmental assessment for part of the so-called downtown relief line soon, a senior city official warned, the project will be put on hold – and it won't be revisited until next year.

So begins another chapter in a familiar transit saga about a relief line that is not currently funded but is almost universally accepted as the most important transit project for the city. Although politicians occasionally use other names for this line to make it more palatable to residents of outlying areas, few dispute the need to take pressure off the Yonge subway line and the Bloor-Yonge interchange, both of which are struggling with demand.

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To avoid another delay, and have the environmental assessment begin in June, councillors will have to agree on funding at committee in May and again at a full council the following month. The six-month assessment would generate a proposed route and station locations for the eastern portion of the line, recommendations that would be presented to the new council in the first part of 2015.

The leading mayoral candidates are generally agreed on the importance of a relief line, though Rob Ford says it has to come after subways under Finch and Sheppard. Olivia Chow, who has been criticized by opponents for her stance, says it is important but too long-term a project to be an issue in this election. John Tory says that this line and the Scarborough subway extension are his "highest priority."

Different ways to funnel people into the core have been debated since long before the current subway system was built. As far back as 1910, three buried streetcar lines into the downtown, one from the east, were at the centre of a plan pitched under the slogan "Tubes for the people." Its proponent lost a mayoral election and the man who took office killed the plan, according to a book on Toronto's transit history.

A century later, the relief line being discussed is a roughly U-shaped pair of subway lines coming into the core from points to the west and east of Yonge. Although the exact routes have not been determined, building even just the eastern portion as far north as Bloor is expected to have a significant effect. A city report said it would cut demand on the Yonge subway line by 12 per cent and reduce transfers at Bloor-Yonge by 30 per cent.

Because of the greater ridership potential on the eastern portion, that leg is the priority and was the subject of a meeting Saturday that allowed the public to weigh in on the process. There will be three more meetings this week, two in Toronto and one in Richmond Hill.

At Saturday's meeting, city planning staff laid out the rationale for the eastern leg to Bloor, the cost of which is pegged at $3.2-billion, in a very preliminary estimate. At the same meeting, the provincial transit agency Metrolinx talked about its work studying the many other ways to reduce transit crowding across the region. These include improved surface rail and would complement a tunnelled downtown relief line.

"We all know this is a critical project for the city of Toronto," said chief planner Jennifer Keesmaat, adding optimistically: "If we don't go sideways in the next year, two years, we're not that far off."

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She cited her young son and said that she looked forward to him attending a ribbon-cutting on the subway line within 15 years.

That will sound ambitious to the Torontonians who have seen transit projects repeatedly changed or derailed as the political winds changed. And while there's a growing sense of momentum around this line, some participants at Saturday's meeting voiced frustration at the pace of the process.

"It feels like we're having a consultation about how to have a consultation," grumbled one participant, who did not want to be identified because he does work with the city.

Indeed, the process can seem obsessively inclusive. At Saturday's meeting, participants were asked to offer insight into how to improve the project's terms of reference and whether there was anything missing from the study process. Meatier issues such as the route the line should take or where stations would be located are still far off.