The party’s Shadow foreign secretary David Lammy claimed the current rules were “bureaucratic” with a “lot of red tape” amid on ongoing row over the numbers able to come to Britain.
His comments add to the intense criticism facing the UK Government over its failure so far to relax the visa requirements for Ukrainian nationals.
The debate has seen figures from the Westminster parties argue on Twitter, with a Tory minister tweeting and deleting a suggestion that Ukraine refugees could apply to pick fruit on British farms.
Mr Lammy said: “It’s insisting that people demonstrate salaries, that they have family ties in this country.
“People are fleeing with their children in their arms. Why would you ask people how rich they are to enter our country? Of course, there are some people who may not have family ties, but want to come into this country.
“No, we should have a scheme and a process similar to the scheme that we had after the Balkans.
"That’s as generous as the schemes that we’ve had in the past, when we allowed people to flee Idi Amin from Uganda, when we allowed people to flee Cyprus, when we allowed the Vietnamese boat people to enter our country.
“That’s the sort of generosity that the British people expect. Frankly, suggesting that people should use this scheme that effectively fruit pickers come to this country on is totally, totally unacceptable.”
It comes as Kevin Foster, who serves under Priti Patel in the Home Office, dismissed criticism of UK Government policy from the Labour MP Luke Pollard.
Replying to a tweet, Mr Foster wrote: “Hi Luke, As you will be well aware there are a number of routes, not least our seasonal worker scheme you will recall from your Shadow DEFRA days, which Ukrainians can qualify for, alongside the family route for those with relatives here.”
The Foreign Secretary Liz Truss declined on Sunday to say how many refugees from Ukraine the UK will accept but insisted the Government is “urgently” looking at what more it can do.
Asked about waiving visa requirements, the senior minister instead focused on the support the UK Government was giving.
She said: “It is a desperate situation. We’re working with the United Nations, we’re working with the Red Cross, to keep humanitarian corridors open.
“We’ve put support into the neighbouring countries like Slovakia and Poland to help with the refugee crisis.
“And of course Britain has always welcomed refugees fleeing from war, and we’re urgently looking at what more we can do to facilitate that.
“But ultimately what we need to make sure is that we protect Ukraine as a sovereign democracy.
“Ultimately, the people of Ukraine want to live in Ukraine.”
It was put to the Cabinet minister that anyone who has no connection to the UK cannot get in, and she was asked if that will be changed.
She said: “We are looking urgently at what we can do. We’re working with European partners about how we support refugees who are leaving Ukraine. So, yes, is the answer.”
Pressed on how many refugees the UK will accept and how soon, Ms Truss said: “Well, as I’ve said, this is something we are urgently looking at."
The United Nations High Commissioner for Refugees has updated its estimate on the number of refugees that have left Ukraine, saying the total stands at 368,000.
It said in a tweet: “Ukraine refugee numbers have just been refreshed – these are based on data made available by national authorities. The current total is now 368,000 and continues to rise.”
Earlier, Wales First Minister Mark Drakeford said the UK will need to “go beyond” its current visa arrangements for those fleeing conflict in Ukraine.
On Saturday night, Prime Minister Boris Johnson said the UK is “of course” going to take refugees and pledged to help people “fleeing in fear of their lives”.
Enver Solomon, CEO of The Refugee Council, said: “Events over the weekend are creating yet more confusion and distress to Ukrainian families desperately seeking reassurance and support from the UK Government regarding their future safety.
“We urgently need the Government to announce a clear plan which immediately relaxes visa requirements to allow family members of Ukrainians in the UK to join them here.
“And a safe route so people at risk of persecution can apply for a humanitarian visa to travel to the UK and claim asylum once here.
“We must uphold our tradition of supporting people fleeing war and bloodshed, and send a clear signal to Ukrainian families that they are welcome in the UK.”
Sunday morning also saw Mr Lammy demand Russia’s ambassador to the UK leave the country after telling “lie after lie”.
It came after Labour leader Sir Keir Starmer said the Government should order Andrei Kelin to leave the country immediately.
Speaking to Sky on Trevor Phillips on Sunday, Mr Lammy said: “He (Mr Kelin) came to Parliament a few weeks ago and we asked him what is Russia doing with troops on the border? He denied that Russia was building troops.
“Has Russia got plans to come into the country? He said ‘no, they do not want to invade Ukraine’.
“It’s time for him to leave our country.”
In a wide ranging interview, Mr Lammy also said more major Russian banks should be included within the UK’s sanctions on the country.