Kompany was sidelined for two months after suffering a calf injury during an FA Cup tie at Stoke in January.
Kidd revealed the 27-year-old missed the final two games of the season because of the niggling problem but his return in City's post-season friendly against Chelsea in St Louis has provided cause for optimism.
"After the FA Cup final he was feeling his calf and he didn't do any training," Kidd, who took the helm after Roberto Mancini was sacked, said.
"In the Reading game, we were desperate to get a point to finish second - it would have been a travesty if the lads hadn't done that, so that was paramount - but Vinny had the problem with the calf and couldn't make it.
"Vinny was obviously wanting to play the last game of the season but for one game it wasn't worth the risk."
Another centre-back, Matija Nastasic, is also still struggling with the problem that kept has kept him out since the FA Cup final loss to Wigan.
"Matija didn't travel," Kidd added. "We left him back in New York to have treatment."
City play Chelsea again in the second of two matches at Yankees Stadium in New York on Saturday.