Russian officials have called it a “terrorist attack” but have not commented on Islamic State's claims.
According to Russia's Ministry of Emergency Situations, about 107 people were still in hospital as of Saturday morning.
Russian Telegram channels including Baza, which is close to the security services, and a lawmaker said some of the suspects were from the Central Asian nation of Tajikistan.
Tajikistan's Foreign Ministry said in a statement that it had not received any information from Moscow about the involvement of its citizens.
Residents in Moscow formed long lines in the rain Saturday morning to donate blood, according to videos posted by state media outlets.
Advertising billboards at some bus stops in Moscow have been replaced by memorial posters featuring a candle, the RIA Novosti state agency reports.
Major events were canceled across the country, including a friendly football match between Russia and Paraguay in Moscow on Monday.
Statements of condemnation kept coming from world leaders.
On Saturday, a spokesman for Afghanistan's Foreign Ministry said the Taliban “strongly condemns the recent terrorist attack in Moscow, Russia… and considers it a gross violation of all humanitarian standards.”