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In June, many Google users were surprised to see an unusual greeting at the top of their Gmail inbox, Google home page or Chrome browser. “Warning: We believe state-sponsored attackers may be attempting to compromise your account or computer.”

On Tuesday, tens of thousands more Google users will begin to see that message. The company said that since it started alerting users to malicious — probably state-sponsored — activity on their computers in June, it has picked up thousands more instances of cyberattacks than it anticipated.

Mike Wiacek, a manager on Google’s information security team, said in an interview on Tuesday that since Google started to alert users to state-sponsored attacks three months ago, it had gathered new intelligence about attack methods and the groups deploying them. He said the company was using that information to warn “tens of thousands of new users” that they may have been targets, starting on Tuesday.

By Tuesday afternoon, several people — many of them American journalists and foreign policy experts — had already taken to Twitter to say they had seen the warning. Noah Schactman, the editor of Wired’s national security blog “Danger Room,” tweeted: “Aaaaand I just got Google’s ‘you may be a victim of a state-sponsored attack’ notice. #WhatTookYouSoLong?” Daveed Gartenstein-Ross, a senior fellow at the Foundation for Defense of Democracies, also reported getting the message. As did Joshua Foust, a fellow at the American Security Project, a nonprofit research organization, who has written extensively about Afghanistan.

Mr. Wiacek noted that Google had seen an increase in state-sponsored activity coming from the Middle East. He declined to call out particular countries, but he said the activity was coming from “a slew of different countries” in the region.

Those findings triangulate with recent discoveries by security researchers that Middle Eastern states, including Iran, Qatar, the United Arab Emirates and Bahrain, have used spyware to monitor citizens and activists overseas.

Last week, several American banks were hit with cyberattacks by hackers claiming Middle Eastern ties. Security researchers have said they have noticed an increase in cyberattacks originating in the region. “We absolutely have seen more activity from the Middle East, and in particular Iran has been increasingly active as they build up their cybercapabilities,” George Kurtz, the president of CrowdStrike, a computer security company, said in a recent interview.

Mr. Wiacek said there were several steps Google users, especially those who get its warning, could take to protect themselves, like changing their e-mail and account passwords, enabling Google’s two-step authentication service and running their computer software updates.