Coppell, 57, is being linked with a return to the Crystal Palace hotseat for a fifth time, following Dougie Freedman's departure to Bolton.
The Reds have no formal contract in place to keep Coppell at the club, but Barker hopes he will carry on his good work at the League One outfit.
"He is so enthusiastic about what we are doing here," Barker told the Crawley Observer. "When we chat at half-time, he means what he is saying. He is so enthusiastic about getting us to the next level.
"I know that he is held in high esteem at Palace but I hope he will bear in mind how well we get on and that might play a part in him staying."