Navalny, 44, launched a hunger strike on 31 March and his medical team over the weekend warned that his health was failing so rapidly he could die at "any minute".
US National Security Adviser Jake Sullivan said on Sunday that Washington had told Moscow "there will be consequences" if Navalny dies in prison, but did not mention specifics.
Concerns over Navalny's health have mounted against the background of a spike in tensions between Moscow and the West over a litany of issues, including Russia's troop build-up on Ukraine's border, interference in United States elections and other perceived hostile activities.
European Union foreign ministers will discuss the case of Navalny when they hold talks on Monday, Germany said.
His lawyers and allies are demanding he be transferred to a regular hospital, but the Kremlin has said Navalny is not entitled to special treatment.
The Kremlin dismissed the outcry from foreign leaders over Mr. Navalny's condition. "The state of health of those convicted and jailed on Russian territory can not and should not be a theme of their interest", spokesman Dmitry Peskov said.More news: Navalny to be transferred to hospital for convicts
The EU in October sanctioned six Russian officials over the nerve agent attack, and in February sanctioned another four individuals over his arrest and sentencing.
Earlier, his French counterpart Jean-Yves Le Drian also said the bloc was watching Navalny's situation closely, as he warned of possible further sanctions against Russian Federation.
Following the U.S. sanctions, Russia recalled its ambassador to the U.S., Anatoly Antonov, and Russian Foreign Minister Sergey Lavrov publicly suggested that the U.S. should likewise recall Sullivan.
Sullivan said he would be returning to Moscow "in the coming weeks before any meeting between Presidents Biden and Putin".
Navalny's team has called on supporters to take to the streets across the country on Wednesday - when Putin is due to deliver his annual state of the nation address - to protest how he is being treated.
Russia has additionally provoked much consternation in the USA and other Western countries with a military buildup on the border to eastern Ukraine, where Russian-backed separatists have been fighting government forces since 2014.More news: Premiers Being Asked To Provide COVID Support To Ontario
Kiev has been battling Russian-backed separatists in eastern Ukraine since 2014 and clashes intensified early this year, effectively shredding a ceasefire agreed last July.
Borrell said that more than 150,000 Russian troops were now massed along Ukraine's borders and in the annexed Crimea peninsula.
Lithuania's Foreign Minister Gabrielius Landsbergis pushed further, saying the bloc should prepare "a humanitarian mission" to fly him out of Russian Federation for treatment.
Washington and Moscow have recently expelled 10 of each other's diplomats in tit-for-tat moves.
Czech authorities accused them of involvement in a deadly 2014 explosion on its soil at a military ammunition warehouse that killed two people.More news: Capitol Police Officer Brian Sicknick Died Of Natural Causes