Batting coach Grant Flower and data analyst G T Niroshan tested positive for the new coronavirus soon after their return from England on Tuesday, Sri Lanka Cricket (SLC) said in a statement on Friday.
The three one-day internationals will now be played on July 18, 20 and 23, followed by three Twenty20 matches on July 25, 27 and 29. All matches will be held in Colombo.
"We understand that the circumstances are extraordinary, but the BCCI would like to extend its full support to Sri Lanka Cricket during these tough times for the smooth conduct of the upcoming series," BCCI secretary Jay Shah said.
"Our medical team is in constant touch with the team of doctors at SLC and together following all safety protocols that will help the series to get underway."
SLC chief executive Ashley de Silva said he was "grateful to the BCCI for understanding the situation".
Three English players and four staff members also tested positive for COVID-19 on Monday, forcing them to field an entirely new squad for their home series against Pakistan.
The SLC on Friday also postponed the Lanka Premier League, which was to begin on July 30, to November-December citing cricket's packed calendar.