The minister is not believed to have the virus herself, but is in self-isolation as a precaution.
Small and Medium Enterprises minister Sithembiso Nyoni faced reporters at the post-cabinet briefing in Mutsvangwa's absence.
ZimLive understands Christopher Mutsvangwa, the minister's husband, tested positive for the virus last week.Government spokesman Ndavaningi Mangwana had not responded to a request for a comment.
Several ministers missed Tuesday's meeting after coming into contact with Perrance Shiri, the lands and agriculture minister who died from the virus last week.
Zimbabwe has reported 4,339 coronavirus cases and 84 deaths since the virus was first detected in the country on March 20.
On Wednesday, the ruling Zanu PF party said 26 of its 154 workers at the party headquarters in Harare had tested positive for the virus which causes the respiratory disease, Covid-19.
"Those who are symptomatic have been advised to seek immediate and urgent medical attention. The party HQ is being fumigated by health officers and members of staff have been asked to stay home during this process," Zanu PF spokesman Patrick Chinamasa said.
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