Irene Coleman (85) claims she was nearly knocked down by a cyclist last Friday evening (March 24) in the precinct.
She says the male rider did not ring a bell to warn her that he was coming and he missed her by just one foot.
Ms Coleman from Milngavie says this is not the first time this has happened to her.
She says cyclists aren’t dismounting from their bikes as they are meant to and often they are going really fast.
She said: “I fear for my safety now when I go shopping.
“Last Friday I had a very near miss, and it’s happened again since then.
“If I was knocked over at my age and broke a hip I would be disabled for the rest of my life.
“I’m not the only person who feels like this - lots of other elderly people are terrified to walk in the precinct.
“It’s also very dangerous for young children who are often running around near the playpark.
“This is certainly not something that we want to hand down to future older people and children.”
Ms Coleman, a former secondary school art teacher, believes that cyclists should be banned from the precinct because they ignore the signs which ask them to dismount.
She added: “There is no full-time police presence to deal with cyclists who flout the rules.
“I often avoid the precinct now - this was the only independent pleasure I had and I resent having this taken away from me.
“I have always enjoyed shopping there, meeting friends, and I need to do my banking and collect medicine from the pharmacy in the precinct.
“Now it’s not a pleasurable experience for me - it makes me feel very anxious.
“If something isn’t done soon there will be some very bad accidents.”
Thomas Glen, depute chief executive - place, neighbourhood & corporate assets at the council, said: “Enforcement of irresponsible cycling in Milngavie precinct is a Police Scotland matter.
“Police officers will be out and about educating cyclists on legislation - including raising awareness at Milngavie BID’s Classic Car Show.
“In addition, we will be looking at signage to ensure that cyclists know to dismount when they enter the precinct.”
Chief Inspector Gerry Corrigan, said: “We are aware of the concerns raised by local residents regarding cyclists not dismounting from their bikes in Milngavie Precinct.
“As the precinct is busy with pedestrians, we would ask that cyclists follow the signs and dismount from their bikes for the short distance.
“We will continue to work with East Dunbartonshire Council to engage and educate cyclists in the area on this safety concern.”