Rwanda are Arsenal's sleeve sponsor, with the “Visit Rwanda” logo printed on the left sleeve of all the Gunners' shirts.
Disappointed after Friday's 2-0 defeat by Brentford in their season opener, Kagame called for changes at the club.
“We just must NOT excuse or Accept mediocrity. A team has to be built with purpose to win win win. So that when we lose...it was not to be expected! I am sure we all know on whose shoulders the heaviest burden rests. I hope they know too or even accept it!!!”
Kagame, who has enjoyed support from Western donors for restoring Rwanda to stability after a 1994 genocide, is also accused by rights groups of being an autocrat and quashing political opposition to extend his 21-year presidency.
Arsenal, who have not qualified for any European football this season, next face London rivals and Champions League holders Chelsea at home on August 22.