Inverness chairman Kenny Cameron says the club "are quietly confident of new arrivals" as they seek to rebuild.
Greg Tansey left the Highlanders for Stevenage while Jonny Hayes and Gregory Tade are also expected to exit.
"We intend to have most of the squad - all if we can - in place for pre-season training on June 29," said Cameron.
"As far as our out-of-contract players are concerned, we have made the best offers possible in the club's current financial position."
David Proctor has been released and manager Terry Butcher has set other players a deadline to accept new contracts.
Ryan Esson is the only one to confirm he is staying but back-up goalkeeper Jonny Tuffey has decided to move on.
"Terry and [assistant] Maurice Malpas have been working hard on prospective targets as we wait to see if our remaining out-of-contract players accept their new offers," Cameron added on the Inverness website.
"The number of new faces we require will depend on how many of our out-of-contract players decide to leave. But our management team are pulling out all the stops to ensure we have adequate cover for all positions.
"I cannot say too much at this stage as potential incoming players are still under contract elsewhere, but several of them have responded in a very positive manner to our initial approaches.
"In light of the changes to the SPL rules, it is imperative that we cut our cloth to suit our income as we move forward."
Butcher said of Tuffey: "We're grateful to Jonny for his service and we appreciate why he has moved on.
"We have a replacement in mind and we are hopeful that we can tie him up fairly soon."