One force said the move was "a precaution".
The company is working with policeOfficers from Hertfordshire, where the firm has its headquarters, said no-one had been hurt.
They said the closures affected stores in Lancashire, Suffolk, Dyfed Powys in Wales, Fife in Scotland, Leicestershire, Humberside, West Mercia, West Yorkshire, Nottinghamshire, Strathclyde and London.
A Hertfordshire police spokeswoman said: "Police are investigating a series of incidents in a number of locations across the country involving Tesco stores.
"These stores have been temporarily closed as a precaution for public safety."
A spokesman for South Wales police said the Tesco store at Riverside, Port Talbot, had been closed since this afternoon, adding: "Officers will maintain a presence at the site for as long as is necessary."
A Tesco spokeswoman said: "Tesco is working with the police following a series of incidents.
"Police are investigating and so we are unable to comment further."
She added: "As always, at all times customer welfare is our priority."