The BBC is reporting officials at the Ukrainian consulate in Edinburgh said their phones had been "ringing off the hook" with offers.
It comes after Ukraine's President Volodymyr Zelensky announced the creation of an international legion to help defend his country and appealed to foreign volunteers to come forward.
Both the UK and Scottish Government have warned against anyone other than trained military personnel heading to the country.
There is also a question mark over the legal status of anyone who travels there to fight.
Across the UK there have been reports of Britons signing up to fight alongside Ukrainians.
A steady stream of people, some with military experience and others who have never seen action, arrived at the Ukrainian embassy in west London on Tuesday to volunteer to fight.
Liam Hawkins, 30, drove to the Ukrainian embassy from Deptford, south east London, to sign up to help refugees stuck at the border in a humanitarian capacity.
The 30-year-old father told reporters: “I haven’t slept since this invasion and I just can’t stay at home with my babies and my missus and do nothing.
“Right is right and wrong is wrong, regardless of nationality or creed or whatever.”
Defence Secretary Ben Wallace urged people not to travel to Ukraine to join the fighting.
Mr Wallace said he does not “want to see British people killed any more than I want to see Ukrainians” dying after Cabinet colleague Liz Truss said she would “absolutely” support British nationals who chose to go to help fight against the Russian invasion.
Mr Wallace, who served in the Scots Guards, said there are “better ways” for Britons to help.
He said: “If you’re keen to help and you’re a United Kingdom citizen, come and join our armed forces.
“Look, there are people who will go… I think what I would say is unless you are properly trained, unless you are a, you know, experienced member of an armed forces, I think there are better ways for you to contribute to the security of Ukraine.”
He said this could be by donating money through the Ukrainian embassy to be spent on weaponry and aid, or to volunteer with organisations helping refugees.