The Nigerian international is on the verge of joining Southampton in a deal which could net Rangers around £10million with a sell-on clause on top.
Aribo is into the final year of his deal at Ibrox, as are Kent, Bassey and Morelos.
McCoist revealed on talkSPORT he was “a little bit concerned” with their current situation and would feel a lot more comfortable with the trio tied down to new deals.
"I would be a lot happier with my Rangers supporting scarf on if we tied them down for another couple of years.
"The one player towards the end of the season who was absolutely outstanding was Calvin Bassey.
"He can play left back or left centre back in a 4 or a 3. He had an incredible man of the match performance in the Europa League final by a considerable distance, then another man of the match performance in the domestic cup final.
"So I am a little bit concerned but there are a lot of really good things happening at Rangers as well."
McCoist also praised the impact Aribo had at Ibrox.
"It will be a sad loss but that is football," he said. "We bought him from Charlton a few years back and I think he has been fabulous.
"He has blossomed and he has grown. He has 20 odd caps for Nigeria, playing top class football.
"If he goes to Southampton they will be getting a very good player."
He added: "Anyway, bottom line is I would have absolutely no problems, with all the Rangers fans, saying thank you to Joe Aribo for his time at Rangers and wishing him all the best.
"He gave us that moment in the Europa League final and he has had some great performances.”
Meanwhile, Rangers are closing in on the signing of striker Antonio Colak from PAOK. The deal is expected to go through in the coming days with the player linking up with his new team-mates in Portugal.
The club have also been linked with a move for DR Congo winger Theo Bongonda.