Border Reivers coach Steve Bates has warned that a decision must be reached soon on the future of the club.
Bates fears a player exodus if the future remains uncertain
Representatives from the Scottish Rugby Union and Scottish Borders Council described Tuesday's talks as constructive and will meet again.
And Bates told BBC Sport: "There is a general feeling from players that they would like to stay.
"But, as time progresses, they will have to look after themselves and make sure they have contracts next year."
The SRU has said it wants to close the Reivers and focus its finances on the Glasgow Warriors professional side.
Supporters have been urged to attend the Reivers home game on Friday evening to show their opposition to the plan.
The game's ruling body has made it clear that if a credible funding package can be found - without SRU finance - the Borders' side could be retained.
The Borders Rugby Action Group claims it can support the team without any backing from the SRU.
And more talks are being planned in the region to look at ways of taking those proposals forward.
Meanwhile, fans have been urged to back the team by turning out at the Magners League game against Ulster on Friday evening at Netherdale.
Border Reivers player and Scotland International Gregor Townsend said the team needed a demonstration of support.
"We know there is support for the Reivers in the Borders and throughout the UK and further afield," he said.
"What we're looking for on Friday is for local people to turn out in large numbers - bring your friends and family.
"It'll be a great atmosphere and a real boost to the players and to everyone involved in trying to keep the team alive."