The scheme will create a segregated pathway between Queen's Park and the city centre.
Design proposals for the final leg of the £6.5m project will be on display at the drop-in event.
The final phase of the 3km route will link the junction at the north end of Victoria Bridge, Bridgegate and King Street.
Staff from Glasgow City Council and Sustrans will be on hand at the event to answer questions about the scheme.
Councillor Richardson said: "The South City Way is already gearing up to become one of the most significant stretches of cycling infrastructure in Scotland. The South City Way will provide a direct cycle route from a major residential area of Glasgow right into the city centre that will ensure safe cycling every inch of the way.
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"The route is intended to improve the cycling experience for those who already use a bike. But it is also hoped that the cycle way will remove the understandable concerns people have about using a bike to get about a busy city. We want more people to regard cycling as a mode of transport they can rely on day after day for the good of our health and environment.
"The South City Way is a key part of our plans to increase cycling numbers in Glasgow. We want people to get involved with the discussion about how our plans are implemented. The drop-in at the Briggait is a great opportunity for people to talk to those directly involved in the project and share their views."
The drop-in at the Briggait is on Thursday 31 January from 11am to 7pm.
