Cllr Neil Heritage, the leader of the Yorkshire Coast Independents Alliance, has submitted the motion that will now be debated at the next meeting of the full council on Wednesday next week.
Cllr Heritage, himself a former serviceman, has been backed in his motion by Cllr Bill Chatt of the Cluster of Independent Members.
In his motion Cllr Heritage, who was elected in 2019 as a UKIP member but later left the party, says the flag should fly above buildings “where appropriate”.
He says it would be a show of unity following Brexit.
The motion states: “All council-owned buildings within the Borough of Scarborough, where appropriate, should fly the Union Jack flag, for example town halls.
“The Union Jack is a symbol of freedom and pride which we should promote for our residents of the borough.
“As an ex-serviceman who has served my country with pride, the Union Jack is a symbol I and we should always respect.
“The Union Jack is a flag representing the United Kingdom of England, Wales, Scotland and Northern Ireland, a true flag of unity.
It is now time to show unity after leaving the European Union whether you agree or not with the democratic outcome of the referendum.”
The motion will be debated by the council and a majority of the 46 councillors would be required to support it in order for it to pass.
Last month, the Government announced that Union Flags would be flown every day on all UK Government buildings.
Before then it was only flown on certain occasions.