The animals are said to have targeted the technician as he walked through the campus of Meerut medical college in Meerut, in Uttar Pradesh state, on Friday.
The group managed to snatch the blood samples of four patients, who have tested positive for Covid-19, before fleeing.
Video footage shared on Twitter shows a monkey sitting in a tree with what appears to be one of the stolen kits.
The animal can be seeing dropping something, which looks to be a white disposable glove, to the ground before chewing on another.
Authorities have said it is not known whether the monkeys have spilled the blood samples. However, people living near the campus fear the virus could be spread to nearby residential areas.
Dr S K Garg, a top official at the college, said: “Monkeys grabbed and fled with the blood samples of four Covid-19 patients who are undergoing treatment ... we had to take their blood samples again.”
He added that it was unclear whether the monkeys could contract the novel coronavirus if they came into contact with infected blood.
“No evidence has been found that monkeys can contract the infection,” he said.
— Sanjay Jha (@JhaSanjay07)May 29, 2020
The virus is believed to have jumped from animals into people in a wildlife market in the Chinese city of Wuhan late last year.
India has had 165,799 cases of the coronavirus and 4,706 deaths.
Monkeys have been increasingly straying into human settlements in India and causing disturbances, even attacking people.
Environmentalists say the destruction of natural habitat is the main reason the animals move into urban areas in search of food.
Additional reporting by Reuters