Their message came as Russian clubs and national teams were suspended from all competitions.
The move comes after the International Olympic Committee (IOC) recommended that Russian and Belarusian athletes and officials are banned from any organised international competitions.
It means the Russian men’s team will not play their World Cup play-off matches next month and the women’s team have been banned from this summer’s European Championship.
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Spartak Moscow have also been kicked out of the Europa League, while Uefa has ended its sponsorship with Russian energy giant Gazprom.
"Both presidents hope that the situation in Ukraine will improve significantly and rapidly so that football can again be a vector for unity and peace amongst people."
Earlier in the day, SFA president Rod Petrie sent a message of support to Ukraine following the invasion by Russia.
Scotland are due to face Ukraine in the World Cup play-off later this month, while the women’s side are scheduled to play the country in a World Cup qualifier in April.
Scotland are in the same group as Russia in the UEFA Regions’ Cup, which is an international amateur tournament, with a game to take place in August. The SFA’s position is that they will not nominate any team at any level to compete against Russia.
A statement said: “The Scottish FA president, Rod Petrie, has written to his counterpart at the Ukrainian Association of Football to send a message of support, friendship and unity.
“Football is inconsequential amid conflict but we have conveyed the strong sense of solidarity communicated to us by Scotland fans and citizens in recent days.
"We remain in dialogue with UEFA and FIFA regarding our men's FIFA World Cup play-off and women’s World Cup qualifier and have offered to support our Ukrainian colleagues’ preparations as best we can in these unimaginably difficult circumstances.
"Should the current circumstances continue, we will not sanction the nomination of a team to participate in our scheduled UEFA Regions Cup fixture against Russia, due to be played in August.
“This will remain our position should any other fixtures arise at any level of international football.”
The SFA decisions decisions followed similar moves made by the English and Welsh FAs in stating they will not play Russia for the foreseeable future.
Scotland captains Andy Robertson and Rachel Corsie expressed their solidarity with Ukraine's footballers ahead of their scheduled matches in the coming weeks.
Scotland are due to host Ukraine at Hamden in a World Cup play-off semi-final on March 24 and the two nations were scheduled to meet in a women's World Cup qualifier in the eastern European country on April 8.
Robertson and Corsie declared that their respective teams "stand in solidarity with our fellow participants from Ukraine's football community".
In a joint statement, they added: "A match, no matter how significant on the pitch, feels unimportant right now, but we intend to express our friendship and unity when we face each other next month.
"In the meantime we pray for your wellbeing and the safety of all those people needlessly suffering. Until we meet, we send our love."