La Toison d’Or will be closed “until further notice”, while The Cavalier says it plans to reopen after three days.
The Golden Fleece said a client had “decided to go out at night” – despite symptoms of Covid-19 – while awaiting test results.
The pub, which was made aware of the situation on Saturday, said the customer was later confirmed to have tested positive.
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The Golden Fleece announced on Facebook that it will remain closed while staff are tested and the pub is thoroughly cleaned.
He said customers who were at the pub on Friday will be contacted via NHS Test and Trace.
A statement on their Facebook page said: “Unfortunately, we will not be open until further notice because an irresponsible customer who was showing symptoms of Covid-19 (while awaiting his test results) decided to go out at night.
“Today we have just learned that they have tested positive. Sorry for any inconvenience caused, but the safety of our customers comes first.
“As soon as our staff have been tested and the pub has been thoroughly cleaned, we will keep you all updated.”
The Cavalier’s Facebook page added: “We were informed this morning that someone who may have had Covid-19 could have been in the bar last night.
“We are trying to locate this person and as a precaution we have closed the bar and restaurant for 72 hours for a thorough cleaning of the bar.
“However, our delivery service is still available from tomorrow as usual by phone and Just Eat.
“Even though that person was away from the kitchen or the staff, as a precaution we are closing deliveries today in order to deep clean the deep kitchen as well.
“Sorry for any inconvenience caused. “
Anyone with reservations for Saturday or Sunday August 2 should contact the pub directly, the statement added.