Navalny, 44, a prominent opponent of President Vladimir Putin, announced a hunger strike last week in protest at what he said was the refusal of prison authorities to treat him properly for acute back and leg pain.
A group of his allies said they would protest at the prison in the town of Pokrov 100 km (60 miles) east of Moscow from Tuesday unless he saw a doctor of his choice and was given what they regarded as proper medicine.
Prison authorities say his condition is satisfactory and he has been provided with all necessary medical care.
Late on Monday, his allies said the protest would go ahead after Navalny said he was continuing his hunger strike, although he had a high temperature and bad cough and three inmates in his ward had been hospitalized with tuberculosis.