The defender suffered what appeared to be a groin injury during City's 4-0 Premier League defeat of Newcastle at the Etihad Stadium and left the field after 71 minutes.
It was the only sour note in what was an otherwise excellent first match under new manager Manuel Pellegrini.
But Kompany, 27, tweeted: "Team was great. I felt great. Injury is a blow after all the hard work. I'll be back to that level in no time. Thanks for all the support!."
City are currently light on central defenders with Matija Nastasic also out with an ankle problem, although Pellegrini did suggest the Serbian would be back in full training this week.
With Micah Richards also missing the first game, midfielder Javi Garcia came on to play alongside Joleon Lescott in a makeshift back line for the closing minutes.
City's only other centre-back option is Dedryck Boyata, who lacked opportunities under former boss Roberto Mancini and spent time on loan at Bolton and FC Twente.
City have been linked with other defenders, including Real Madrid's Pepe, but Pellegrini has given no firm indication that he intends to move back into the transfer market.
Kompany's presence is vital to the City team and a two-month absence with a calf injury in the second half of last season proved costly in the title race.
City will now await the results of a scan on the Belgium skipper.
Pellegrini said: "We will see what happened with the injury. The doctor will say how days or weeks he will be injured.
"I can't tell you exactly now what happened with him but I think it is a serious injury - maybe for one or two weeks."
City travel to Cardiff next weekend and then host Hull before the September international break.
Kompany's fellow defender Pablo Zabaleta said: "It is hard for the team to lose the captain.
"He is one of the most important players for the team but sometimes it happens in football.
"Obviously after the pre-season people are a little bit tired. I wish him a speedy recovery and hopefully he can come back as soon as possible."