The warning came as world leaders gathered in Brussels for a Nato summit, at which concerns about Russia look set to dominate.
Ahead of the meeting the Nato Secretary-General Jens Stoltenberg said relations with Russia were at the "lowest point" since the end of the Cold War.
Mr Johnson said that although he was hopeful the UK’s relationship with Russia could improve, more than three years after the Salisbury poisonings, progress thus far had been “pretty disappointing”.
He added: “Nato allies stood by Britain then. I know that President Biden will be taking some pretty tough messages to President Putin in the course of the next few days.”
The prime minister also insisted that Nato did not want a new Cold War with China, although he acknowledged that he superpower posed "challenges" for the Western nuclear alliance.
President Biden is due to meet Mr Putin for talks in Geneva on Wednesday.
Mr Johnson is expected to use the Nato summit to highlight the way the Covid crisis has exacerbated a number of security threats - including cyber attacks.
Earlier Mr Stoltenberg warned such attacks formed part of a pattern of Russian behaviour that Nato allies could not afford to ignore.
"We see attempts to meddle in our political democratic processes, to undermine the trust in our institutions and efforts to divide us. We have to take that very seriously," he told Times Radio.
"We need to strengthen our cyber defences, we need to exchange intelligence, we need to be vigilant and aware of all these different tools of aggressive actions, military and non-military."
Elsewhere, the Defence Secretary Ben Wallace suggested former prime minister Theresa May would make an "excellent" candidate to succeed Mr Stoltenberg when he steps down next year.
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