The former Who Wants to Be A Millionaire? host claimed the government scheme for offering shelter to those fleeing the warzone had been a "nightmare of civil service bureaucratic c**p", but claimed he "sort of managed to short-cut all that."
Tarrant, 75, told The Sun: “We just look after them and help them with shopping and stuff."
He added: "They are lovely, and it makes you think - they’ve got a baby, and these poor people have come all the way across Ukraine to get out of that bloody hellhole.
"They keep talking about going home some time, but I say, ‘You don’t even know what your home’s like. Is it a pile of bricks? You don’t know what’s there waiting for you’. So they are very welcome to stay.”
The TV and radio presenter - who has four adult children - lives in Berkshire with partner Jane Bird, and they have welcomed a mother and baby, as well as her elderly mother to live with them. They cannot be named for security reasons and the husband remains in Ukraine fighting in the war.
Tarrant said he was put in touch with the family when they were sleeping on the floor of an airport in Poland and managed to organise the paperwork for them to be granted asylum in the UK.
He said he is not accepting the £350 monthly payment offered by the government as part of the Homes for Ukraine scheme.
Tarrant said: "The baby speaks nothing - just Ukrainian gurgles, I think. The mum has reasonable English and her mum doesn’t speak much at all. But they’re just really nice people.”
Russian president Vladimir Putin began his military invasion of Ukraine on 24 February. Cities in the country have suffered mass destruction and hundreds of thousands of Ukrainians have fled the country.
The Homes For Ukraine scheme - which is allows UK residents, who have a spare room, to sponsor an individual or family from Ukraine to live with them for up to six months - has been criticised for the slow process in matching refugees with those offering their homes.
Since the government website launched on 18 March over 200,000 UK residents have registered their interest.
And while government figures show that 40,000 UK visas have been issued, just 6,600 Ukrainians have actually arrived.
Lineker has already shared his home with a refugee two years ago.
Watch: Benedict Cumberbatch has offered to house Ukrainian refugees