The bank wants to install the system on the roof ledges at the back of the Manor Street branch, which "delivers a small electric shock to deter pest birds from unwanted areas".
Planning documents say the shock "in no way harms the bird" and works in a similar way to an agricultural electric fence.
The application says the kittiwakes have been nesting and causing a mess, presenting a slip hazard to customers, staff and pedestrians. The guano waste they produce is acidic and can erode the building's fabric.
It comes after the TSB bank on the corner of Beck Hill and Manor Street, had to shut temporarily for repairs in August, after the nests of a herring gull and kittiwake damaged the roof, sparking a major leak.
A specialist contractor will carry out the work at the Barclays branch, which is in the Bridlington Quay conservation area.
It states: "Effective and professional management of vermin will ensure the longevity of the fabric of the heritage asset for years to come."
Barclays has also applied to East Riding Council to put in blunted "points" at the rear of the building, which don't harm the birds but prevents them from alighting.
The building and others on Manor Street already have bird spikes fitted on their historic frontages.
A Scarborough resident who has been using an electronic system on one a property to stop gulls roosting says it delivers a 24-volt shock for a millisecond.
He said: "The birds jump up as they are uncomfortable.
"The problem with spikes is they build nests on them and netting doesn't work because they get behind it.
"It is the most humane way of seagull-proofing the roof. The birds land on it once and don't come back."
A Barclays spokesperson said the proposals would replace torn netting, adding that waste from the kittiwakes had created "a hazard for customers and staff and a risk to the health and safety of other pedestrians".