Benetton's European Challenge Cup game against Agen in Italy on Saturday has been cancelled as has the scheduled clash between Stade Francais and Cardiff Blues.
Tournament organisers European Professional Club Rugby said: "EPCR has been informed of positive Covid-19 tests recorded by members of the Benetton Rugby testing pool.
"As a consequence, the club's Challenge Cup preliminary stage match against Agen, which was scheduled for today at Stadio di Monigo, is cancelled.
"As per its Covid-19 protocol, EPCR will convene a match result resolution committee to determine the result of the cancelled match, and the decision of the committee will be communicated as soon as practicable. "
Agen, beaten at home by London Irish in their Challenge Cup opener last weekend, can realistically expect to be awarded five points from the fixture against their Italian opponents.
Three matches in the Heineken Champions Cup this weekend were cancelled, and one postponed, because of Covid-related issues.
La Rochelle, Lyon and Toulouse were all awarded five points from cancelled games against Bath, Glasgow and Exeter, respectively.
EPCR, meanwhile, postponed Friday's game between the Scarlets ad Toulon after the French club decided not to play following a positive Covid-19 test in the Scarlets squad.
A decision on points allocation from that match has not yet been made.
Stade Francais have also forfeited their Challenge Cup clash with Cardiff Blues over Covid-19 concerns.
Stade raised fears over Coronavirus issues to tournament bosses, revealing their reluctance to travel for Sunday's fixture at Rodney Parade. EPCR awarded Cardiff a 28-0 win, with Stade returning no positive Covid tests.
Stade were left concerned about their Covid status after last week's fixture with Benetton, and the Italians since returning a number of positive Coronavirus tests.
A statement read: "Although there is no evidence that any Stade Francais Paris players or staff were exposed to the virus during the Round 1 fixture, and they have all subsequently tested negative for the virus, there are doubts over the Covid-19 status of the travelling matchday squad and staff, which have prompted the club to report their concerns to EPCR.
"Although a Medical Risk Assessment Committee was satisfied that tomorrow's fixture at Rodney Parade could go ahead safely, Stade Francais Paris were reluctant to travel to Wales.
"EPCR would again like to emphasise that blame was not considered as a factor during the deliberations, and the committee's decision was made with a view to facilitating the completion of this season's Challenge Cup in unprecedented circumstances."