The government has been facing growing calls to waive visa rules for Ukrainians seeking sanctuary in the UK.
Labour says the government's failure to relax visa restrictions for Ukrainians is "immoral".
The UN says about 368,000 refugees have left Ukraine since the conflict began.
The number of refugees is expected to increase as Russian forces attempt to seize control of Ukrainian cities and overthrow the democratically elected government in the capital Kyiv.
The UK government has introduced temporary visa concessions for the family members of British nationals in Ukraine, including waiving the application fees.
But British citizens have told the BBC they have experienced long waits and technical issues when attempting to secure visas for their family members in Ukraine.
In updated guidance, the Home Office said the UK has "multiple official visa routes available for Ukrainians who are not dependants of British nationals".
The guidance says they can apply through a British consulate in the Ukrainian city of Lviv or a neighbouring country, such Hungary, Poland, or Romania.
When asked if the government would waive visa requirements for any Ukrainians coming to the UK, Ms Truss told BBC One's Sunday Morning programme: "It is a desperate situation.
"Of course, Britain has always welcomed refugees fleeing from war and we are urgently looking at what more we can do to facilitate that."
Labour and the SNP have urged the government to do much more to help Ukrainian refugees come to the UK.
They have called on the government to follow the example of the Republic of Ireland, which has waived all visa requirements for all Ukrainian citizens entering the country.
Labour's shadow foreign secretary David Lammy has said visa restrictions for those seeking sanctuary in the UK from Ukraine were "unacceptable".
He told Sunday Morning the government was "asking people to jump through hoops" and called on Home Secretary Priti Patel to "do more" and "act quickly".
"There's bureaucracy, there's red tape," Mr Lammy said. "Why would we ask people what their salary levels are when you're fleeing war? That is immoral. It is not in the traditions of this country."
He referred to a now-deleted tweet by immigration minister Kevin Foster, who said "there are number of routes" those wanting to move to the UK can take, "not least our Seasonal Worker Scheme".
Under the scheme, foreign nationals can apply for temporary visas to work on farms picking fruit and vegetables among other jobs in the UK.
Shadow home secretary Yvette Cooper said the immigration minister was "right" to delete the tweet, Ukrainians were still facing "high hurdles and delays" to visas, she added.
Wales's First Minister Mark Drakeford has said the UK will need to "go beyond" its current visa arrangements for those fleeing conflict in Ukraine.
On Saturday Prime Minister Boris Johnson said "of course we are going to take refugees", adding "this country has had a historic and proud role of taking refugees from all conflicts."
A period of anxiety awaits Dr Nataliya Rumyantseva and her Ukrainian mother, Valentyna Klymova, who is attempting to apply for a visa to enter the UK.
So far, it's been a "confusing" process said Dr Rumyantseva, a Ukrainian citizen with permanent residence in the UK.
Fearing an imminent Russian invasion, Ms Klymova left her hometown of Kharkiv and on 24 February crossed the border to Hungary on foot.
From there the 69-year-old, who doesn't speak English, managed to hitch a ride to Budapest and board a flight to Paris.
Dr Rumyantseva met her mother in Paris on Saturday. She tried to cross the border from France to the UK claiming asylum, but was denied entry.
Her options, border officials told her, were either to seek asylum in France as it's a safe country, or apply for a standard visitor visa to the UK.
Dr Rumyantseva's partner, British national Nick Jeal, said they paid €394 (£331; $444) plus a €120 appointment fee to apply for the visa.
"There is no mechanism to allow anyone in the country at the moment," he said. "We can't get her in the country without an extended visa process."
That process, he said, was "not possible for someone fleeing their bombed country".
Dr Andrii Zharikov also has family and friends in Ukraine, including his 55-year-old father, who has volunteered for the Ukrainian army to fight against Russian forces.
He told BBC Radio 5 Live his younger sister, 19, had fled to western Ukraine but was now struggling to leave the country.
"The UK government has not eased any visa restrictions," said Dr Zharikov, a senior lecturer at the University of Portsmouth.
"I cannot get her here. So at best, I can only hope to get her across the border to one of the EU countries."
He urged the UK government to relax visa rules, not only for British nationals who have family in Ukraine, but also for Ukrainians nationals.
"We have the means to support them, we just need to ease the process of getting people to their immediate family," said Dr Zharikov.