Main Street Bridgend on the Derry/Donegal border
He wrote to Stephen Donnelly on Wednesday to make the request.
The letter states that both governments should be doing “all we can to prevent non-essential cross-border travel”, and “clear messaging” and enforcement should be used, if required.
It comes after seven cases of Indian variant were found in the North of Ireland.
Mr Swann has called on the Irish government to elicit an “appropriate and proportionate” response to recent transmission data on both sides of the border.
There are concerns about the high level of transmission in Donegal which borders Derry
“Our two jurisdictions are at different junctures, in terms of number of cases, the current trajectory of the epidemic, vaccination progress and Covid-19 cases in respective jurisdictions,” the letter says.
“In the present circumstances, crossing the border for non-essential shopping or socialising purposes creates an unnecessary risk of virus spread.”
It is understood the issue was raised when Stormont ministers met on Thursday to discuss the coronavirus restrictions in Northern Ireland.
Mr Swann has asked to meet Mr Donnelly, along with other senior departmental officials and health agencies to consider extra measures to control infection rates in border areas.
He said he looked forward to a time when normal cross-border activity could resume, but warned: “We are not there yet.”
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