Law firm Mishcon de Reya is representing the group of small business policyholders known as the Hiscox Action Group. The law firm has written to Hiscox to trigger arbitration clauses in nearly 350 contracts, the action group said in a statement.
Lloyd's of London insurer Hiscox said it welcomed "all proposals to expedite resolution of any dispute."
Britain's coronavirus lockdown has triggered insurance claims from companies seeking compensation for having to shut down their activities.
Hiscox and other insurers have said that most contracts do not cover the pandemic but they aim to pay valid claims quickly.
Industry regulator, the Financial Conduct Authority, is in the process of seeking clarity from the courts on whether the wording of some insurance policies should provide cover during the pandemic.
The arbitration claim by the Hiscox Action Group will take place alongside the FCA's test case against eight insurers, including Hiscox, the group said.
Small businesses in France and the United States are also fighting insurers over policies which did not pay out following government lockdowns due to the virus.