Both were residents at Lennel House Care Centre, where 24 people - 12 residents and 12 members of staff - have now tested positive for the virus.
Scottish Borders Council contracts a number of places at the home, which is run by St Philips Care.
The authority said that referrals to Lennel House had been suspended.
A spokesperson for Scottish Borders Council said: “We are working alongside NHS Borders, the Care Inspectorate and the management of Lennel House following concerns that have been raised with regards to the care and safety of residents.
“Enquiries are at an early stage but we are liaising closely with all partners and providing support to management, staff, residents and families."
The council was alerted over safety concerns following the contained outbreak at the home, which it says is ongoing.
An Incident Management Team is being led by the public health team at NHS Borders, with representation from SB Cares and Lennel House as well as the council, which says it will ensure a “robust action plan” is developed and taken forward.
The spokesperson added: "We fully appreciate this is a difficult and worrying time for residents, their families and staff, and we would like to reassure them that we are doing all we can to support everyone.
“Residents have been tested and enhanced testing has been introduced for staff members. Everyone who has been confirmed to have the infection, and their close contacts, have been identified and received appropriate advice.”
A spokesperson for St Philips Care said that staff, who were being supported with additional resources, were “working tirelessly” to support residents at a time which is “extremely stressful” for everyone involved.
They added: “We work very closely with the Scottish Borders Health & Social Care Partnership (Scottish Borders Council and NHS Borders) who has supported us to ensure every precaution is taken to protect our residents.”
Most of the residents are believed to have received the first dose of their Covid-19 vaccination in December.