Appleton hopes to meet Chainrai to reach an agreement to start bringing new players in.
Chainrai and the Pompey Supporters Trust are bidding for control of the Fratton Park side, which has eight days to avoid closure.
The three senior players still at the club - Dave Kitson, Tal Ben Haim and Liam Lawrence - must leave or accept a wage reduction before August 10 for Pompey to stave off liquidation.
"I think if the three remaining senior players left then obviously I would be given the green light to move straight away and sign players," Appleton told BBC Radio Solent. "But I'd like to think over the next few days I would be given the opportunity to speak to Portpin [Chainrai's company] and come up with a different solution.
He added: "We've got a few players in on trial who are waiting for me to be given the green light so I can sign them. There are certain players in there I would sign very quickly.
"You've got to give them a little bit of credit for being so patient as they are desperate to get a contract. Next Friday is looming and we want to be in a position where we can get on with things."