The city council's cabinet member for transport, councillor Lynne Stagg will be asked to approve a new March end date on Thursday with the possibility that this is pushed even further back to November 2022.
Cllr Stagg said the scheme has been 'successful' in helping reduce car journeys and said operator Voi had been 'responsive' in dealing with incidents.
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Almost 30,000 people have used the rental scheme since it started March with each scooter being used more than twice a day on average. A survey found more than half of these journeys replaced either a car or taxi trip.
But concerns have been raised about their safety more than 100 accidents recorded since March with more than 13 deemed to be 'serious' having required hospital treatment.
Voi said it was difficult to verify many of the complaints with users often not following up on issues they raise.
Portchester teenager Jack Martin collided with an e-scooter whose rider, he said, suddenly turned off to his right without warning. He said there were still issues that needed to be resolved before it was expanded.
‘I’m not happy because I still see people riding like idiots,' he said. 'I have been put off them.
‘I think it’s bad they carry on but if they are here to stay then it will not bother me. They should have more guidelines and have someone at the stands where you rent them saying what you can and can’t do. They should only issue them to people with good intentions.'
Cllr Stagg, who in August criticised Voi's handling of underage riders, acknowledged there some concerns with the scheme still remained but said the firm had been 'responsive' to criticism and had brought in extra safety measures.
'I think it's been quite successful so far and I know a lot of residents won't agree with me on that but a lot of that comes down to the confusion with privately-owned scooters which are a real issue,' she said.
'If they are ridden properly they are a great asset for our city - they are an effective alternative to car journeys, helping to deal with our air pollution problem. Of course it's not going to solve it on its own but every little helps.
'There is a problem with private scooters and we have had issues but generally the trial has been a success and that's great - we want it to be a success.'
She said she was particularly concerned there could be an influx of these given as Christmas gifts leading to issues in the coming months.
Privately-owned e-scooters cannot be insured, making their use on public land or on roads illegal.