It denies plans to invade but has well over 100,000 troops near the border.
The prime ministers trip is part of a surge of diplomatic activity, with Foreign Secretary Liz Truss and Defence Secretary Ben Wallace also due to meet their Russian counterparts in Moscow on Thursday.
Ahead of the first visit to Russia by a UK foreign secretary in four years, Ms Truss said she was determined to stand up for freedom and democracy in Ukraine and intended to urge Moscow to pursue a diplomatic solution.
Labour leader Sir Keir Starmer is also expected to meet the Nato secretary-general.
The latest diplomatic efforts come as Russia and Belarus prepare to start 10 days of joint military drills, which the US described as escalatory.
Meanwhile, Frances President Emmanuel Macron said talks with Russia and Ukraine, which began earlier this week, could resume on Thursday.
Mr Macron said that President Vladimir Putin had assured him Russian forces would not ramp up the crisis, but Russia said it had given no such guarantee.
The current tensions come eight years after Russia annexed Ukraine’s Southern Crimea peninsula.
Since then, Ukraine’s military has been locked in a war with Russian-backed rebels in areas of the east near Russia’s borders.
The conflict has claimed 14,000 lives and caused at least two million people to flee their homes,
A direct threat to security Nato’s 30 members which include the UK, US and several former Soviet republics agree to come to anothers aid in the event of an armed attack.
Ukraine is not a member of Nato.
However, it is a partner country and may be allowed to join the alliance sometime in the future.
Russia sees this as a direct threat to its security, arguing that Nato’s expansion eastward breaks a promise made by the US in 1990, though interpretations differ over exactly what was said.
Some analysts believe the current build-up of Russian troops on the Ukrainian border may be an attempt to force the West to take Russia’s security demands seriously.
Mr Johnson will meet the Nato secretary-general in Brussels and Poland’s president and prime minister in Warsaw, intending to show solidarity with Nato allies who are perceived to be most at risk from Russian hostilities.
The prime minister announced 1,000 more British troops will be put on alert in the UK, with the government warning of a possible humanitarian
crisis, if a Russian incursion into Ukraine sends people fleeing into neighbouring countries; he has stressed that Nato was unwilling to send troops into Ukraine itself.
Some 350 Royal Marines from 45 Commando are to begin deploying to Poland as part of support announced for the country earlier this week.
The UK will also offer to double the number of its troops in Estonia, deploy more RAF jets based in southern Europe, and to sail a Type 45 destroyer and HMS Trent, a patrol ship, in the eastern Mediterranean.
British PM to meet NATO allies amid Ukraine tensions