A total of 1,130 players and club personnel were tested in the fourth screening session, which took place on Thursday and Friday and yielded the first all-clear.
There were six positive cases from three clubs in round one, two from two clubs in round two and four from two clubs in round three.
The latest results have given a boost to the prospects of the top-flight getting back under way on June 17 - a fortnight on Wednesday.
Two matches take place on that day, including the Blades' trip to Aston Villa.
Osborn said: "It was obviously good news with the number of positive tests thankfully not being too high in the Premier League from the start.
"We were all expecting - especially with the first set of results - to be quite high as no-one had a clue really if they'd got it or not.
"But especially with the latest set of results, where no-one has got it, it definitely does reassure you."
Meanwhile, the former Nottingham Forest player has stressed that he and his colleagues believe that the game's return in a fortnight's time is a realistic time-frame and is not too early.
Osborn added: "I do feel ready. We had a rigourous programme when we were sent home in lockdown and we only had a week off.
"Other than that, we have been keeping as fit as we can do and the sessions have been pretty tough - three or four sessions a day and then as soon as we have been allowed back in the training ground, we have been in at every stage just gearing.
"It has been just like pre-season really - even harder. We are just starting to get some minutes in behind closed doors games between ourselves.
"We are gearing up the minutes. It was fifty minutes the other day and seventy today and next week, we will have a couple more games to try and get the intensity and fitness back."