There were no comprehensive reports of turnout throughout the country and it was unclear if the demonstrations would match the size and intensity of nationwide protests that broke out in January after Navalny, President Vladimir Putin's most prominent opponent, was arrested.
On Sunday, National Security AdvisorJake Sullivan told CNN's State of the Union that "we have communicated to the Russian government that what happens to Mr. Navalny in their custody is their responsibility and they will be held accountable by the global community".
Navalny's supporters have called for what they hope will be the biggest protests in modern Russian history. She was taken to a police station and charged with organising an illegal gathering.
Two close aides were detained by police in Moscow, while monitors reported police raids on Navalny's offices and arrests of his supporters across the country.
Sobol was removed from a taxi by uniformed policei, said her lawyer, Vladimir Voronin.
Navalny is being held in prison on a two-and-a-half year fraud sentence and has gone on a three-week hunger strike to demand he receive medical care from his own doctors - after back pain and loss of feeling in his legs - while officials say he is already getting adequate treatment.
Navalny was detained when he returned to Russian Federation in January after months recovering in Germany from a near-fatal poisoning he blames on the Kremlin - an accusation it rejects.
He was detained for what Russian officials said was him breaking probation by being hospitalized in Germany after surviving a poisoning previous year.
Navalny, who survived a nerve agent poisoning last year that Russian Federation denies carrying out, was sentenced in February to 2-1/2 years for parole violations related to an embezzlement case that he says was politically motivated.
Navalny's physician, Dr. Yaroslav Ashikhmin, said recently that test results he received from Navalny's family showed sharply elevated levels of potassium, which can bring on cardiac arrest, and heightened creatinine levels that indicate impaired kidneys and mean that he "could die at any moment".
Russia's prison service insisted his condition was "satisfactory", despite moving him to the medical facility, and said he was taking vitamin supplements as part of his treatment.
His organisation had said protests would take place in more than 180 cities, but it was not immediately clear if they would match the massive turnout for protests in January that were the largest in Russian Federation in a decade. Prison officials rebuffed attempts by his doctors to visit him there.
"Russian authorities are responsible for Aleksey Navalny's well-being", Ned Price, a spokesman for the State Department, tweeted Sunday.
Navalny's allies vowed to continue their work despite the pressure. On Friday, state prosecutors in Moscow said they wanted to label his regional groups and anti-corruption foundation "extremist", a move that would essentially outlaw their activity.
"It is, of course, an element of escalation", Ashurkov told the AP. "We have neither the intention nor the possibility to abandon what we're doing".